Calculus Examples

$f (x) = \frac{5}{9} (4 x - 1) (x + 2) (x - 3)$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $\frac{5}{9}$ is constant with respect to $x$ , the derivative of $\frac{5}{9} (4 x - 1) (x + 2) (x - 3)$ with respect to $x$ is $\frac{5}{9} \frac{d}{d x} [((4 x - 1) (x + 2)) (x - 3)]$ .

$f' (x) = \frac{5}{9} \cdot \frac{d}{d x} (((4 x - 1) (x + 2)) (x - 3))$

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = (4 x - 1) (x + 2)$ and $g (x) = x - 3$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) \frac{d}{d x} (x - 3) + (x - 3) \frac{d}{d x} ((4 x - 1) (x + 2)))$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} ((4 x - 1) (x + 2)))$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) (1 + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} ((4 x - 1) (x + 2)))$

Step 1.1.3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) (1 + 0) + (x - 3) \frac{d}{d x} ((4 x - 1) (x + 2)))$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $1$ and $0$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) \cdot 1 + (x - 3) \frac{d}{d x} ((4 x - 1) (x + 2)))$

Step 1.1.3.4.2

Multiply $4 x - 1$ by $1$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) \frac{d}{d x} ((4 x - 1) (x + 2)))$

Step 1.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 4 x - 1$ and $g (x) = x + 2$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) ((4 x - 1) \frac{d}{d x} (x + 2) + (x + 2) \frac{d}{d x} (4 x - 1)))$

Step 1.1.5

Differentiate.

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Step 1.1.5.1

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) ((4 x - 1) (\frac{d}{d x} (x) + \frac{d}{d x} (2)) + (x + 2) \frac{d}{d x} (4 x - 1)))$

Step 1.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) ((4 x - 1) (1 + \frac{d}{d x} (2)) + (x + 2) \frac{d}{d x} (4 x - 1)))$

Step 1.1.5.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) ((4 x - 1) (1 + 0) + (x + 2) \frac{d}{d x} (4 x - 1)))$

Step 1.1.5.4

Simplify the expression.

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Step 1.1.5.4.1

Add $1$ and $0$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) ((4 x - 1) \cdot 1 + (x + 2) \frac{d}{d x} (4 x - 1)))$

Step 1.1.5.4.2

Multiply $4 x - 1$ by $1$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) \frac{d}{d x} (4 x - 1)))$

Step 1.1.5.5

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) (\frac{d}{d x} (4 x) + \frac{d}{d x} (- 1))))$

Step 1.1.5.6

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) (4 \frac{d}{d x} (x) + \frac{d}{d x} (- 1))))$

Step 1.1.5.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) (4 \cdot 1 + \frac{d}{d x} (- 1))))$

Step 1.1.5.8

Multiply $4$ by $1$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) (4 + \frac{d}{d x} (- 1))))$

Step 1.1.5.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) (4 + 0)))$

Step 1.1.5.10

Simplify the expression.

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Step 1.1.5.10.1

Add $4$ and $0$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + (x + 2) \cdot 4))$

Step 1.1.5.10.2

Move $4$ to the left of $x + 2$ .

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + 4 \cdot (x + 2)))$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2) + (x - 3) (4 x - 1 + 4 x + 4 \cdot 2))$

Step 1.1.6.2

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) (x + 2)) + \frac{5}{9} \cdot ((x - 3) (4 x - 1 + 4 x + 4 \cdot 2))$

Step 1.1.6.3

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot ((4 x - 1) x + (4 x - 1) \cdot 2) + \frac{5}{9} \cdot ((x - 3) (4 x - 1 + 4 x + 4 \cdot 2))$

Step 1.1.6.4

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x - 1 x + (4 x - 1) \cdot 2) + \frac{5}{9} \cdot ((x - 3) (4 x - 1 + 4 x + 4 \cdot 2))$

Step 1.1.6.5

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x - 1 x + 4 x \cdot 2 - 1 \cdot 2) + \frac{5}{9} \cdot ((x - 3) (4 x - 1 + 4 x + 4 \cdot 2))$

Step 1.1.6.6

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot ((x - 3) (4 x - 1 + 4 x + 4 \cdot 2))$

Step 1.1.6.7

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot ((x - 3) (4 x) + (x - 3) \cdot - 1 + (x - 3) (4 x) + (x - 3) (4 \cdot 2))$

Step 1.1.6.8

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x) - 3 (4 x) + (x - 3) \cdot - 1 + (x - 3) (4 x) + (x - 3) (4 \cdot 2))$

Step 1.1.6.9

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x) - 3 (4 x) + x \cdot - 1 - 3 \cdot - 1 + (x - 3) (4 x) + (x - 3) (4 \cdot 2))$

Step 1.1.6.10

Apply the distributive property.

Step 1.1.6.11

Apply the distributive property.

Step 1.1.6.12

Apply the distributive property.

$f' (x) = \frac{5}{9} \cdot (4 x \cdot x) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13

Combine terms.

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Step 1.1.6.13.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{5}{9} \cdot (4 (x \cdot x)) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.2

Raise $x$ to the power of $1$ .

Step 1.1.6.13.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{5}{9} \cdot (4 x^{1 + 1}) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.4

Add $1$ and $1$ .

$f' (x) = \frac{5}{9} \cdot (4 x^{2}) + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.5

Combine $4$ and $\frac{5}{9}$ .

$f' (x) = \frac{4 \cdot 5}{9} x^{2} + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.6

Multiply $4$ by $5$ .

$f' (x) = \frac{20}{9} x^{2} + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.7

Combine $\frac{20}{9}$ and $x^{2}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{5}{9} \cdot (- 1 x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.8

Rewrite $- 1 x$ as $- x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{5}{9} \cdot (- x) + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.9

Combine $\frac{5}{9}$ and $x$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{5}{9} \cdot (4 x \cdot 2) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.10

Multiply $2$ by $4$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{5}{9} \cdot (8 x) + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.11

Combine $8$ and $\frac{5}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{8 \cdot 5}{9} x + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.12

Multiply $8$ by $5$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{40}{9} x + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.13

Combine $\frac{40}{9}$ and $x$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{40 x}{9} + \frac{5}{9} \cdot (- 1 \cdot 2) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.14

Multiply $- 1$ by $2$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{40 x}{9} + \frac{5}{9} \cdot - 2 + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.15

Combine $\frac{5}{9}$ and $- 2$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{40 x}{9} + \frac{5 \cdot - 2}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.16

Multiply $5$ by $- 2$ .

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{40 x}{9} + \frac{- 10}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.17

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} - \frac{5 x}{9} + \frac{40 x}{9} - \frac{10}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.18

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{9} + \frac{- 5 x + 40 x}{9} - \frac{10}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.19

Add $- 5 x$ and $40 x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.20

Raise $x$ to the power of $1$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{5}{9} \cdot (4 (x \cdot x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.21

Raise $x$ to the power of $1$ .

Step 1.1.6.13.22

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{5}{9} \cdot (4 x^{1 + 1}) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.23

Add $1$ and $1$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{5}{9} \cdot (4 x^{2}) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.24

Combine $4$ and $\frac{5}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{4 \cdot 5}{9} x^{2} + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.25

Multiply $4$ by $5$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20}{9} x^{2} + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.26

Combine $\frac{20}{9}$ and $x^{2}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.27

Multiply $4$ by $- 3$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{5}{9} \cdot (- 12 x) + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.28

Combine $- 12$ and $\frac{5}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 12 \cdot 5}{9} x + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.29

Multiply $- 12$ by $5$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 60}{9} x + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.30

Combine $\frac{- 60}{9}$ and $x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 60 x}{9} + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.31

Cancel the common factor of $- 60$ and $9$ .

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Step 1.1.6.13.31.1

Factor $3$ out of $- 60 x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{3 (- 20 x)}{9} + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.31.2

Cancel the common factors.

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Step 1.1.6.13.31.2.1

Factor $3$ out of $9$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{3 (- 20 x)}{3 (3)} + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.31.2.2

Cancel the common factor.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{3 (- 20 x)}{3 \cdot 3} + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.31.2.3

Rewrite the expression.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 20 x}{3} + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.32

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{(20) x}{3} + \frac{5}{9} \cdot (x \cdot - 1) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.33

Move $- 1$ to the left of $x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{5}{9} \cdot (- 1 \cdot x) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.34

Rewrite $- 1 x$ as $- x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{5}{9} \cdot (- x) + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.35

Combine $\frac{5}{9}$ and $x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{5}{9} \cdot (- 3 \cdot - 1) + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.36

Multiply $- 3$ by $- 1$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{5}{9} \cdot 3 + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.37

Combine $\frac{5}{9}$ and $3$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{5 \cdot 3}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.38

Multiply $5$ by $3$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{15}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.39

Cancel the common factor of $15$ and $9$ .

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Step 1.1.6.13.39.1

Factor $3$ out of $15$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{3 (5)}{9} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.39.2

Cancel the common factors.

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Step 1.1.6.13.39.2.1

Factor $3$ out of $9$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{3 \cdot 5}{3 \cdot 3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.39.2.2

Cancel the common factor.

Step 1.1.6.13.39.2.3

Rewrite the expression.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} - \frac{5 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.40

To write $- \frac{20 x}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x}{3} \cdot \frac{3}{3} - \frac{5 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.41

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.6.13.41.1

Multiply $\frac{20 x}{3}$ by $\frac{3}{3}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x \cdot 3}{3 \cdot 3} - \frac{5 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.41.2

Multiply $3$ by $3$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{20 x \cdot 3}{9} - \frac{5 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.42

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 20 x \cdot 3 - 5 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.43

Multiply $3$ by $- 20$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 60 x - 5 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.44

Subtract $5 x$ from $- 60 x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} + \frac{- 65 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.45

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{(65) x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (x (4 x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.46

Raise $x$ to the power of $1$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (4 (x \cdot x)) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.47

Raise $x$ to the power of $1$ .

Step 1.1.6.13.48

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (4 x^{1 + 1}) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.49

Add $1$ and $1$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{5}{9} \cdot (4 x^{2}) + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.50

Combine $4$ and $\frac{5}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{4 \cdot 5}{9} x^{2} + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.51

Multiply $4$ by $5$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20}{9} x^{2} + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.52

Combine $\frac{20}{9}$ and $x^{2}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{5}{9} \cdot (- 3 (4 x)) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.53

Multiply $4$ by $- 3$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{5}{9} \cdot (- 12 x) + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.54

Combine $- 12$ and $\frac{5}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 12 \cdot 5}{9} x + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.55

Multiply $- 12$ by $5$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 60}{9} x + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.56

Combine $\frac{- 60}{9}$ and $x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 60 x}{9} + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.57

Cancel the common factor of $- 60$ and $9$ .

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Step 1.1.6.13.57.1

Factor $3$ out of $- 60 x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{3 (- 20 x)}{9} + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.57.2

Cancel the common factors.

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Step 1.1.6.13.57.2.1

Factor $3$ out of $9$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{3 (- 20 x)}{3 (3)} + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.57.2.2

Cancel the common factor.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{3 (- 20 x)}{3 \cdot 3} + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.57.2.3

Rewrite the expression.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 20 x}{3} + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.58

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{(20) x}{3} + \frac{5}{9} \cdot (x (4 \cdot 2)) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.59

Multiply $4$ by $2$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{5}{9} \cdot (x \cdot 8) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.60

Move $8$ to the left of $x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{5}{9} \cdot (8 \cdot x) + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.61

Combine $8$ and $\frac{5}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{8 \cdot 5}{9} x + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.62

Multiply $8$ by $5$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40}{9} x + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.63

Combine $\frac{40}{9}$ and $x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{5}{9} \cdot (- 3 (4 \cdot 2))$

Step 1.1.6.13.64

Multiply $4$ by $2$ .

Step 1.1.6.13.65

Multiply $- 3$ by $8$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{5}{9} \cdot - 24$

Step 1.1.6.13.66

Combine $\frac{5}{9}$ and $- 24$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{5 \cdot - 24}{9}$

Step 1.1.6.13.67

Multiply $5$ by $- 24$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{- 120}{9}$

Step 1.1.6.13.68

Cancel the common factor of $- 120$ and $9$ .

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Step 1.1.6.13.68.1

Factor $3$ out of $- 120$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{3 (- 40)}{9}$

Step 1.1.6.13.68.2

Cancel the common factors.

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Step 1.1.6.13.68.2.1

Factor $3$ out of $9$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{3 \cdot - 40}{3 \cdot 3}$

Step 1.1.6.13.68.2.2

Cancel the common factor.

Step 1.1.6.13.68.2.3

Rewrite the expression.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} + \frac{- 40}{3}$

Step 1.1.6.13.69

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} + \frac{40 x}{9} - \frac{40}{3}$

Step 1.1.6.13.70

To write $- \frac{20 x}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x}{3} \cdot \frac{3}{3} + \frac{40 x}{9} - \frac{40}{3}$

Step 1.1.6.13.71

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.6.13.71.1

Multiply $\frac{20 x}{3}$ by $\frac{3}{3}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x \cdot 3}{3 \cdot 3} + \frac{40 x}{9} - \frac{40}{3}$

Step 1.1.6.13.71.2

Multiply $3$ by $3$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{20 x \cdot 3}{9} + \frac{40 x}{9} - \frac{40}{3}$

Step 1.1.6.13.72

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 20 x \cdot 3 + 40 x}{9} - \frac{40}{3}$

Step 1.1.6.13.73

Multiply $3$ by $- 20$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 60 x + 40 x}{9} - \frac{40}{3}$

Step 1.1.6.13.74

Add $- 60 x$ and $40 x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} + \frac{- 20 x}{9} - \frac{40}{3}$

Step 1.1.6.13.75

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{20 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} + \frac{20 x^{2}}{9} - \frac{(20) x}{9} - \frac{40}{3}$

Step 1.1.6.13.76

Add $\frac{20 x^{2}}{9}$ and $\frac{20 x^{2}}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + 2 (\frac{20 x^{2}}{9}) - \frac{65 x}{9} + \frac{5}{3} - \frac{20 x}{9} - \frac{40}{3}$

Step 1.1.6.13.77

Combine $2$ and $\frac{20 x^{2}}{9}$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{2 (20 x^{2})}{9} - \frac{65 x}{9} + \frac{5}{3} - \frac{20 x}{9} - \frac{40}{3}$

Step 1.1.6.13.78

Multiply $20$ by $2$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} - \frac{65 x}{9} + \frac{5}{3} - \frac{20 x}{9} - \frac{40}{3}$

Step 1.1.6.13.79

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} + \frac{- 65 x - 20 x}{9} + \frac{5}{3} - \frac{40}{3}$

Step 1.1.6.13.80

Subtract $20 x$ from $- 65 x$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} + \frac{- 85 x}{9} + \frac{5}{3} - \frac{40}{3}$

Step 1.1.6.13.81

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} - \frac{(85) x}{9} + \frac{5}{3} - \frac{40}{3}$

Step 1.1.6.13.82

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} - \frac{85 x}{9} + \frac{5 - 40}{3}$

Step 1.1.6.13.83

Subtract $40$ from $5$ .

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} - \frac{85 x}{9} + \frac{- 35}{3}$

Step 1.1.6.13.84

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} + \frac{40 x^{2}}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.85

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2} + 40 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.86

Add $20 x^{2}$ and $40 x^{2}$ .

$f' (x) = \frac{60 x^{2}}{9} + \frac{35 x}{9} - \frac{10}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.87

Cancel the common factor of $60$ and $9$ .

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Step 1.1.6.13.87.1

Factor $3$ out of $60 x^{2}$ .

$f' (x) = \frac{3 (20 x^{2})}{9} + \frac{35 x}{9} - \frac{10}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.87.2

Cancel the common factors.

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Step 1.1.6.13.87.2.1

Factor $3$ out of $9$ .

$f' (x) = \frac{3 (20 x^{2})}{3 (3)} + \frac{35 x}{9} - \frac{10}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.87.2.2

Cancel the common factor.

$f' (x) = \frac{3 (20 x^{2})}{3 \cdot 3} + \frac{35 x}{9} - \frac{10}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.87.2.3

Rewrite the expression.

$f' (x) = \frac{20 x^{2}}{3} + \frac{35 x}{9} - \frac{10}{9} - \frac{85 x}{9} - \frac{35}{3}$

Step 1.1.6.13.88

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{3} + \frac{35 x - 85 x}{9} - \frac{10}{9} - \frac{35}{3}$

Step 1.1.6.13.89

Subtract $85 x$ from $35 x$ .

$f' (x) = \frac{20 x^{2}}{3} + \frac{- 50 x}{9} - \frac{10}{9} - \frac{35}{3}$

Step 1.1.6.13.90

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{3} - \frac{(50) x}{9} - \frac{10}{9} - \frac{35}{3}$

Step 1.1.6.13.91

To write $- \frac{35}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{10}{9} - \frac{35}{3} \cdot \frac{3}{3}$

Step 1.1.6.13.92

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.6.13.92.1

Multiply $\frac{35}{3}$ by $\frac{3}{3}$ .

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{10}{9} - \frac{35 \cdot 3}{3 \cdot 3}$

Step 1.1.6.13.92.2

Multiply $3$ by $3$ .

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{10}{9} - \frac{35 \cdot 3}{9}$

Step 1.1.6.13.93

Combine the numerators over the common denominator.

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} + \frac{- 10 - 35 \cdot 3}{9}$

Step 1.1.6.13.94

Simplify the numerator.

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Step 1.1.6.13.94.1

Multiply $- 35$ by $3$ .

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} + \frac{- 10 - 105}{9}$

Step 1.1.6.13.94.2

Subtract $105$ from $- 10$ .

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} + \frac{- 115}{9}$

Step 1.1.6.13.95

Move the negative in front of the fraction.

$f' (x) = \frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9}$ .

$\frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9} = 0$

Step 2.2

Multiply each term in $\frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9} = 0$ by $9$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{20 x^{2}}{3} - \frac{50 x}{9} - \frac{115}{9} = 0$ by $9$ .

$\frac{20 x^{2}}{3} \cdot 9 - \frac{50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1.1

Factor $3$ out of $9$ .

$\frac{20 x^{2}}{3} \cdot (3 (3)) - \frac{50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.1.2

Cancel the common factor.

$\frac{20 x^{2}}{3} \cdot (3 \cdot 3) - \frac{50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.1.3

Rewrite the expression.

$20 x^{2} \cdot 3 - \frac{50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.2

Multiply $3$ by $20$ .

$60 x^{2} - \frac{50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.3

Cancel the common factor of $9$ .

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Step 2.2.2.1.3.1

Move the leading negative in $- \frac{50 x}{9}$ into the numerator.

$60 x^{2} + \frac{- 50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.3.2

Cancel the common factor.

$60 x^{2} + \frac{- 50 x}{9} \cdot 9 - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.3.3

Rewrite the expression.

$60 x^{2} - 50 x - \frac{115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.4

Cancel the common factor of $9$ .

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Step 2.2.2.1.4.1

Move the leading negative in $- \frac{115}{9}$ into the numerator.

$60 x^{2} - 50 x + \frac{- 115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.4.2

Cancel the common factor.

$60 x^{2} - 50 x + \frac{- 115}{9} \cdot 9 = 0 \cdot 9$

Step 2.2.2.1.4.3

Rewrite the expression.

$60 x^{2} - 50 x - 115 = 0 \cdot 9$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $9$ .

$60 x^{2} - 50 x - 115 = 0$

Step 2.3

Factor $5$ out of $60 x^{2} - 50 x - 115$ .

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Step 2.3.1

Factor $5$ out of $60 x^{2}$ .

$5 (12 x^{2}) - 50 x - 115 = 0$

Step 2.3.2

Factor $5$ out of $- 50 x$ .

$5 (12 x^{2}) + 5 (- 10 x) - 115 = 0$

Step 2.3.3

Factor $5$ out of $- 115$ .

$5 (12 x^{2}) + 5 (- 10 x) + 5 (- 23) = 0$

Step 2.3.4

Factor $5$ out of $5 (12 x^{2}) + 5 (- 10 x)$ .

$5 (12 x^{2} - 10 x) + 5 (- 23) = 0$

Step 2.3.5

Factor $5$ out of $5 (12 x^{2} - 10 x) + 5 (- 23)$ .

$5 (12 x^{2} - 10 x - 23) = 0$

Step 2.4

Divide each term in $5 (12 x^{2} - 10 x - 23) = 0$ by $5$ and simplify.

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Step 2.4.1

Divide each term in $5 (12 x^{2} - 10 x - 23) = 0$ by $5$ .

$\frac{5 (12 x^{2} - 10 x - 23)}{5} = \frac{0}{5}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{5 (12 x^{2} - 10 x - 23)}{5} = \frac{0}{5}$

Step 2.4.2.1.2

Divide $12 x^{2} - 10 x - 23$ by $1$ .

$12 x^{2} - 10 x - 23 = \frac{0}{5}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $0$ by $5$ .

$12 x^{2} - 10 x - 23 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.6

Substitute the values $a = 12$ , $b = - 10$ , and $c = - 23$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (12 \cdot - 23)}}{2 \cdot 12}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 12 \cdot - 23}}{2 \cdot 12}$

Step 2.7.1.2

Multiply $- 4 \cdot 12 \cdot - 23$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{10 \pm \sqrt{100 - 48 \cdot - 23}}{2 \cdot 12}$

Step 2.7.1.2.2

Multiply $- 48$ by $- 23$ .

$x = \frac{10 \pm \sqrt{100 + 1104}}{2 \cdot 12}$

Step 2.7.1.3

Add $100$ and $1104$ .

$x = \frac{10 \pm \sqrt{1204}}{2 \cdot 12}$

Step 2.7.1.4

Rewrite $1204$ as $2^{2} \cdot 301$ .

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Step 2.7.1.4.1

Factor $4$ out of $1204$ .

$x = \frac{10 \pm \sqrt{4 (301)}}{2 \cdot 12}$

Step 2.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 301}}{2 \cdot 12}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{301}}{2 \cdot 12}$

Step 2.7.2

Multiply $2$ by $12$ .

$x = \frac{10 \pm 2 \sqrt{301}}{24}$

Step 2.7.3

Simplify $\frac{10 \pm 2 \sqrt{301}}{24}$ .

$x = \frac{5 \pm \sqrt{301}}{12}$

Step 2.8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 12 \cdot - 23}}{2 \cdot 12}$

Step 2.8.1.2

Multiply $- 4 \cdot 12 \cdot - 23$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{10 \pm \sqrt{100 - 48 \cdot - 23}}{2 \cdot 12}$

Step 2.8.1.2.2

Multiply $- 48$ by $- 23$ .

$x = \frac{10 \pm \sqrt{100 + 1104}}{2 \cdot 12}$

Step 2.8.1.3

Add $100$ and $1104$ .

$x = \frac{10 \pm \sqrt{1204}}{2 \cdot 12}$

Step 2.8.1.4

Rewrite $1204$ as $2^{2} \cdot 301$ .

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Step 2.8.1.4.1

Factor $4$ out of $1204$ .

$x = \frac{10 \pm \sqrt{4 (301)}}{2 \cdot 12}$

Step 2.8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 301}}{2 \cdot 12}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{301}}{2 \cdot 12}$

Step 2.8.2

Multiply $2$ by $12$ .

$x = \frac{10 \pm 2 \sqrt{301}}{24}$

Step 2.8.3

Simplify $\frac{10 \pm 2 \sqrt{301}}{24}$ .

$x = \frac{5 \pm \sqrt{301}}{12}$

Step 2.8.4

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{301}}{12}$

Step 2.9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.9.1

Simplify the numerator.

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Step 2.9.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 12 \cdot - 23}}{2 \cdot 12}$

Step 2.9.1.2

Multiply $- 4 \cdot 12 \cdot - 23$ .

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Step 2.9.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{10 \pm \sqrt{100 - 48 \cdot - 23}}{2 \cdot 12}$

Step 2.9.1.2.2

Multiply $- 48$ by $- 23$ .

$x = \frac{10 \pm \sqrt{100 + 1104}}{2 \cdot 12}$

Step 2.9.1.3

Add $100$ and $1104$ .

$x = \frac{10 \pm \sqrt{1204}}{2 \cdot 12}$

Step 2.9.1.4

Rewrite $1204$ as $2^{2} \cdot 301$ .

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Step 2.9.1.4.1

Factor $4$ out of $1204$ .

$x = \frac{10 \pm \sqrt{4 (301)}}{2 \cdot 12}$

Step 2.9.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 301}}{2 \cdot 12}$

Step 2.9.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{301}}{2 \cdot 12}$

Step 2.9.2

Multiply $2$ by $12$ .

$x = \frac{10 \pm 2 \sqrt{301}}{24}$

Step 2.9.3

Simplify $\frac{10 \pm 2 \sqrt{301}}{24}$ .

$x = \frac{5 \pm \sqrt{301}}{12}$

Step 2.9.4

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{301}}{12}$

Step 2.10

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{301}}{12}, \frac{5 - \sqrt{301}}{12}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{5}{9} \cdot ((4 x - 1) (x + 2) (x - 3))$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{5 + \sqrt{301}}{12}$ .

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Step 4.1.1

Substitute $\frac{5 + \sqrt{301}}{12}$ for $x$ .

$\frac{5}{9} \cdot ((4 (\frac{5 + \sqrt{301}}{12}) - 1) ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Factor $4$ out of $12$ .

$\frac{5}{9} \cdot ((4 \frac{5 + \sqrt{301}}{4 (3)} - 1) ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.1.2

Cancel the common factor.

$\frac{5}{9} \cdot ((4 \frac{5 + \sqrt{301}}{4 \cdot 3} - 1) ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.1.3

Rewrite the expression.

$\frac{5}{9} \cdot ((\frac{5 + \sqrt{301}}{3} - 1) ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{9} \cdot ((\frac{5 + \sqrt{301}}{3} - 1 \cdot \frac{3}{3}) ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.3

Combine fractions.

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Step 4.1.2.3.1

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{5}{9} \cdot ((\frac{5 + \sqrt{301}}{3} + \frac{- 1 \cdot 3}{3}) ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.3.2

Combine the numerators over the common denominator.

$\frac{5}{9} \cdot (\frac{5 + \sqrt{301} - 1 \cdot 3}{3} ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.4

Simplify the numerator.

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Step 4.1.2.4.1

Multiply $- 1$ by $3$ .

$\frac{5}{9} \cdot (\frac{5 + \sqrt{301} - 3}{3} ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.4.2

Subtract $3$ from $5$ .

$\frac{5}{9} \cdot (\frac{2 + \sqrt{301}}{3} ((\frac{5 + \sqrt{301}}{12}) + 2) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{2 + \sqrt{301}}{3} (\frac{5 + \sqrt{301}}{12} + 2 \cdot \frac{12}{12}) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.6

Combine fractions.

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Step 4.1.2.6.1

Combine $2$ and $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{2 + \sqrt{301}}{3} (\frac{5 + \sqrt{301}}{12} + \frac{2 \cdot 12}{12}) ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.6.2

Combine the numerators over the common denominator.

$\frac{5}{9} \cdot (\frac{2 + \sqrt{301}}{3} \cdot \frac{5 + \sqrt{301} + 2 \cdot 12}{12} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.7

Simplify the numerator.

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Step 4.1.2.7.1

Multiply $2$ by $12$ .

$\frac{5}{9} \cdot (\frac{2 + \sqrt{301}}{3} \cdot \frac{5 + \sqrt{301} + 24}{12} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.7.2

Add $5$ and $24$ .

$\frac{5}{9} \cdot (\frac{2 + \sqrt{301}}{3} \cdot \frac{29 + \sqrt{301}}{12} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.8

Multiply $\frac{2 + \sqrt{301}}{3} \cdot \frac{29 + \sqrt{301}}{12}$ .

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Step 4.1.2.8.1

Multiply $\frac{2 + \sqrt{301}}{3}$ by $\frac{29 + \sqrt{301}}{12}$ .

$\frac{5}{9} \cdot (\frac{(2 + \sqrt{301}) (29 + \sqrt{301})}{3 \cdot 12} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.8.2

Multiply $3$ by $12$ .

$\frac{5}{9} \cdot (\frac{(2 + \sqrt{301}) (29 + \sqrt{301})}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.9

Expand $(2 + \sqrt{301}) (29 + \sqrt{301})$ using the FOIL Method.

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Step 4.1.2.9.1

Apply the distributive property.

$\frac{5}{9} \cdot (\frac{2 (29 + \sqrt{301}) + \sqrt{301} (29 + \sqrt{301})}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.9.2

Apply the distributive property.

$\frac{5}{9} \cdot (\frac{2 \cdot 29 + 2 \sqrt{301} + \sqrt{301} (29 + \sqrt{301})}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.9.3

Apply the distributive property.

$\frac{5}{9} \cdot (\frac{2 \cdot 29 + 2 \sqrt{301} + \sqrt{301} \cdot 29 + \sqrt{301} \sqrt{301}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10

Simplify and combine like terms.

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Step 4.1.2.10.1

Simplify each term.

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Step 4.1.2.10.1.1

Multiply $2$ by $29$ .

$\frac{5}{9} \cdot (\frac{58 + 2 \sqrt{301} + \sqrt{301} \cdot 29 + \sqrt{301} \sqrt{301}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.1.2

Move $29$ to the left of $\sqrt{301}$ .

$\frac{5}{9} \cdot (\frac{58 + 2 \sqrt{301} + 29 \cdot \sqrt{301} + \sqrt{301} \sqrt{301}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.1.3

Combine using the product rule for radicals.

$\frac{5}{9} \cdot (\frac{58 + 2 \sqrt{301} + 29 \sqrt{301} + \sqrt{301 \cdot 301}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.1.4

Multiply $301$ by $301$ .

$\frac{5}{9} \cdot (\frac{58 + 2 \sqrt{301} + 29 \sqrt{301} + \sqrt{90601}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.1.5

Rewrite $90601$ as $301^{2}$ .

$\frac{5}{9} \cdot (\frac{58 + 2 \sqrt{301} + 29 \sqrt{301} + \sqrt{301^{2}}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{5}{9} \cdot (\frac{58 + 2 \sqrt{301} + 29 \sqrt{301} + 301}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.2

Add $58$ and $301$ .

$\frac{5}{9} \cdot (\frac{359 + 2 \sqrt{301} + 29 \sqrt{301}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.10.3

Add $2 \sqrt{301}$ and $29 \sqrt{301}$ .

$\frac{5}{9} \cdot (\frac{359 + 31 \sqrt{301}}{36} ((\frac{5 + \sqrt{301}}{12}) - 3))$

Step 4.1.2.11

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{359 + 31 \sqrt{301}}{36} (\frac{5 + \sqrt{301}}{12} - 3 \cdot \frac{12}{12}))$

Step 4.1.2.12

Combine fractions.

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Step 4.1.2.12.1

Combine $- 3$ and $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{359 + 31 \sqrt{301}}{36} (\frac{5 + \sqrt{301}}{12} + \frac{- 3 \cdot 12}{12}))$

Step 4.1.2.12.2

Combine the numerators over the common denominator.

$\frac{5}{9} \cdot (\frac{359 + 31 \sqrt{301}}{36} \cdot \frac{5 + \sqrt{301} - 3 \cdot 12}{12})$

Step 4.1.2.13

Simplify the numerator.

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Step 4.1.2.13.1

Multiply $- 3$ by $12$ .

$\frac{5}{9} \cdot (\frac{359 + 31 \sqrt{301}}{36} \cdot \frac{5 + \sqrt{301} - 36}{12})$

Step 4.1.2.13.2

Subtract $36$ from $5$ .

$\frac{5}{9} \cdot (\frac{359 + 31 \sqrt{301}}{36} \cdot \frac{- 31 + \sqrt{301}}{12})$

Step 4.1.2.14

Multiply $\frac{359 + 31 \sqrt{301}}{36} \cdot \frac{- 31 + \sqrt{301}}{12}$ .

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Step 4.1.2.14.1

Multiply $\frac{359 + 31 \sqrt{301}}{36}$ by $\frac{- 31 + \sqrt{301}}{12}$ .

$\frac{5}{9} \cdot \frac{(359 + 31 \sqrt{301}) (- 31 + \sqrt{301})}{36 \cdot 12}$

Step 4.1.2.14.2

Multiply $36$ by $12$ .

$\frac{5}{9} \cdot \frac{(359 + 31 \sqrt{301}) (- 31 + \sqrt{301})}{432}$

Step 4.1.2.15

Expand $(359 + 31 \sqrt{301}) (- 31 + \sqrt{301})$ using the FOIL Method.

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Step 4.1.2.15.1

Apply the distributive property.

$\frac{5}{9} \cdot \frac{359 (- 31 + \sqrt{301}) + 31 \sqrt{301} (- 31 + \sqrt{301})}{432}$

Step 4.1.2.15.2

Apply the distributive property.

$\frac{5}{9} \cdot \frac{359 \cdot - 31 + 359 \sqrt{301} + 31 \sqrt{301} (- 31 + \sqrt{301})}{432}$

Step 4.1.2.15.3

Apply the distributive property.

$\frac{5}{9} \cdot \frac{359 \cdot - 31 + 359 \sqrt{301} + 31 \sqrt{301} \cdot - 31 + 31 \sqrt{301} \sqrt{301}}{432}$

Step 4.1.2.16

Simplify and combine like terms.

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Step 4.1.2.16.1

Simplify each term.

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Step 4.1.2.16.1.1

Multiply $359$ by $- 31$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} + 31 \sqrt{301} \cdot - 31 + 31 \sqrt{301} \sqrt{301}}{432}$

Step 4.1.2.16.1.2

Multiply $- 31$ by $31$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 \sqrt{301} \sqrt{301}}{432}$

Step 4.1.2.16.1.3

Multiply $31 \sqrt{301} \sqrt{301}$ .

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Step 4.1.2.16.1.3.1

Raise $\sqrt{301}$ to the power of $1$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 ({\sqrt{301}}^{1} \sqrt{301})}{432}$

Step 4.1.2.16.1.3.2

Raise $\sqrt{301}$ to the power of $1$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 ({\sqrt{301}}^{1} {\sqrt{301}}^{1})}{432}$

Step 4.1.2.16.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 {\sqrt{301}}^{1 + 1}}{432}$

Step 4.1.2.16.1.3.4

Add $1$ and $1$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 {\sqrt{301}}^{2}}{432}$

Step 4.1.2.16.1.4

Rewrite ${\sqrt{301}}^{2}$ as $301$ .

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Step 4.1.2.16.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{301}$ as $301^{\frac{1}{2}}$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 {(301^{\frac{1}{2}})}^{2}}{432}$

Step 4.1.2.16.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 \cdot 301^{\frac{1}{2} \cdot 2}}{432}$

Step 4.1.2.16.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 \cdot 301^{\frac{2}{2}}}{432}$

Step 4.1.2.16.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.2.16.1.4.4.1

Cancel the common factor.

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 \cdot 301^{\frac{2}{2}}}{432}$

Step 4.1.2.16.1.4.4.2

Rewrite the expression.

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 \cdot 301^{1}}{432}$

Step 4.1.2.16.1.4.5

Evaluate the exponent.

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 31 \cdot 301}{432}$

Step 4.1.2.16.1.5

Multiply $31$ by $301$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 \sqrt{301} - 961 \sqrt{301} + 9331}{432}$

Step 4.1.2.16.2

Add $- 11129$ and $9331$ .

$\frac{5}{9} \cdot \frac{- 1798 + 359 \sqrt{301} - 961 \sqrt{301}}{432}$

Step 4.1.2.16.3

Subtract $961 \sqrt{301}$ from $359 \sqrt{301}$ .

$\frac{5}{9} \cdot \frac{- 1798 - 602 \sqrt{301}}{432}$

Step 4.1.2.17

Cancel the common factor of $- 1798 - 602 \sqrt{301}$ and $432$ .

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Step 4.1.2.17.1

Factor $2$ out of $- 1798$ .

$\frac{5}{9} \cdot \frac{2 (- 899) - 602 \sqrt{301}}{432}$

Step 4.1.2.17.2

Factor $2$ out of $- 602 \sqrt{301}$ .

$\frac{5}{9} \cdot \frac{2 (- 899) + 2 (- 301 \sqrt{301})}{432}$

Step 4.1.2.17.3

Factor $2$ out of $2 (- 899) + 2 (- 301 \sqrt{301})$ .

$\frac{5}{9} \cdot \frac{2 (- 899 - 301 \sqrt{301})}{432}$

Step 4.1.2.17.4

Cancel the common factors.

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Step 4.1.2.17.4.1

Factor $2$ out of $432$ .

$\frac{5}{9} \cdot \frac{2 (- 899 - 301 \sqrt{301})}{2 \cdot 216}$

Step 4.1.2.17.4.2

Cancel the common factor.

$\frac{5}{9} \cdot \frac{2 (- 899 - 301 \sqrt{301})}{2 \cdot 216}$

Step 4.1.2.17.4.3

Rewrite the expression.

$\frac{5}{9} \cdot \frac{- 899 - 301 \sqrt{301}}{216}$

Step 4.1.2.18

Multiply $\frac{5}{9} \cdot \frac{- 899 - 301 \sqrt{301}}{216}$ .

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Step 4.1.2.18.1

Multiply $\frac{5}{9}$ by $\frac{- 899 - 301 \sqrt{301}}{216}$ .

$\frac{5 (- 899 - 301 \sqrt{301})}{9 \cdot 216}$

Step 4.1.2.18.2

Multiply $9$ by $216$ .

$\frac{5 (- 899 - 301 \sqrt{301})}{1944}$

Step 4.1.2.19

Rewrite $- 899$ as $- 1 (899)$ .

$\frac{5 (- 1 (899) - 301 \sqrt{301})}{1944}$

Step 4.1.2.20

Factor $- 1$ out of $- 301 \sqrt{301}$ .

$\frac{5 (- 1 (899) - (301 \sqrt{301}))}{1944}$

Step 4.1.2.21

Factor $- 1$ out of $- 1 (899) - (301 \sqrt{301})$ .

$\frac{5 (- 1 (899 + 301 \sqrt{301}))}{1944}$

Step 4.1.2.22

Move the negative in front of the fraction.

$- \frac{5 (899 + 301 \sqrt{301})}{1944}$

Step 4.2

Evaluate at $x = \frac{5 - \sqrt{301}}{12}$ .

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Step 4.2.1

Substitute $\frac{5 - \sqrt{301}}{12}$ for $x$ .

$\frac{5}{9} \cdot ((4 (\frac{5 - \sqrt{301}}{12}) - 1) ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Factor $4$ out of $12$ .

$\frac{5}{9} \cdot ((4 \frac{5 - \sqrt{301}}{4 (3)} - 1) ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.1.2

Cancel the common factor.

$\frac{5}{9} \cdot ((4 \frac{5 - \sqrt{301}}{4 \cdot 3} - 1) ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.1.3

Rewrite the expression.

$\frac{5}{9} \cdot ((\frac{5 - \sqrt{301}}{3} - 1) ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{9} \cdot ((\frac{5 - \sqrt{301}}{3} - 1 \cdot \frac{3}{3}) ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.3

Combine fractions.

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Step 4.2.2.3.1

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{5}{9} \cdot ((\frac{5 - \sqrt{301}}{3} + \frac{- 1 \cdot 3}{3}) ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.3.2

Combine the numerators over the common denominator.

$\frac{5}{9} \cdot (\frac{5 - \sqrt{301} - 1 \cdot 3}{3} ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.4

Simplify the numerator.

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Step 4.2.2.4.1

Multiply $- 1$ by $3$ .

$\frac{5}{9} \cdot (\frac{5 - \sqrt{301} - 3}{3} ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.4.2

Subtract $3$ from $5$ .

$\frac{5}{9} \cdot (\frac{2 - \sqrt{301}}{3} ((\frac{5 - \sqrt{301}}{12}) + 2) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{2 - \sqrt{301}}{3} (\frac{5 - \sqrt{301}}{12} + 2 \cdot \frac{12}{12}) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.6

Combine fractions.

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Step 4.2.2.6.1

Combine $2$ and $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{2 - \sqrt{301}}{3} (\frac{5 - \sqrt{301}}{12} + \frac{2 \cdot 12}{12}) ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.6.2

Combine the numerators over the common denominator.

$\frac{5}{9} \cdot (\frac{2 - \sqrt{301}}{3} \cdot \frac{5 - \sqrt{301} + 2 \cdot 12}{12} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Multiply $2$ by $12$ .

$\frac{5}{9} \cdot (\frac{2 - \sqrt{301}}{3} \cdot \frac{5 - \sqrt{301} + 24}{12} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.7.2

Add $5$ and $24$ .

$\frac{5}{9} \cdot (\frac{2 - \sqrt{301}}{3} \cdot \frac{29 - \sqrt{301}}{12} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.8

Multiply $\frac{2 - \sqrt{301}}{3} \cdot \frac{29 - \sqrt{301}}{12}$ .

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Step 4.2.2.8.1

Multiply $\frac{2 - \sqrt{301}}{3}$ by $\frac{29 - \sqrt{301}}{12}$ .

$\frac{5}{9} \cdot (\frac{(2 - \sqrt{301}) (29 - \sqrt{301})}{3 \cdot 12} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.8.2

Multiply $3$ by $12$ .

$\frac{5}{9} \cdot (\frac{(2 - \sqrt{301}) (29 - \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.9

Expand $(2 - \sqrt{301}) (29 - \sqrt{301})$ using the FOIL Method.

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Step 4.2.2.9.1

Apply the distributive property.

$\frac{5}{9} \cdot (\frac{2 (29 - \sqrt{301}) - \sqrt{301} (29 - \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.9.2

Apply the distributive property.

$\frac{5}{9} \cdot (\frac{2 \cdot 29 + 2 (- \sqrt{301}) - \sqrt{301} (29 - \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.9.3

Apply the distributive property.

$\frac{5}{9} \cdot (\frac{2 \cdot 29 + 2 (- \sqrt{301}) - \sqrt{301} \cdot 29 - \sqrt{301} (- \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10

Simplify and combine like terms.

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Step 4.2.2.10.1

Simplify each term.

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Step 4.2.2.10.1.1

Multiply $2$ by $29$ .

$\frac{5}{9} \cdot (\frac{58 + 2 (- \sqrt{301}) - \sqrt{301} \cdot 29 - \sqrt{301} (- \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.2

Multiply $- 1$ by $2$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - \sqrt{301} \cdot 29 - \sqrt{301} (- \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.3

Multiply $29$ by $- 1$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} - \sqrt{301} (- \sqrt{301})}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.4

Multiply $- \sqrt{301} (- \sqrt{301})$ .

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Step 4.2.2.10.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + 1 \sqrt{301} \sqrt{301}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.4.2

Multiply $\sqrt{301}$ by $1$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + \sqrt{301} \sqrt{301}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.4.3

Raise $\sqrt{301}$ to the power of $1$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + {\sqrt{301}}^{1} \sqrt{301}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.4.4

Raise $\sqrt{301}$ to the power of $1$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + {\sqrt{301}}^{1} {\sqrt{301}}^{1}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + {\sqrt{301}}^{1 + 1}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.4.6

Add $1$ and $1$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + {\sqrt{301}}^{2}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.5

Rewrite ${\sqrt{301}}^{2}$ as $301$ .

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Step 4.2.2.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{301}$ as $301^{\frac{1}{2}}$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + {(301^{\frac{1}{2}})}^{2}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + 301^{\frac{1}{2} \cdot 2}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + 301^{\frac{2}{2}}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.10.1.5.4.1

Cancel the common factor.

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + 301^{\frac{2}{2}}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.5.4.2

Rewrite the expression.

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + 301^{1}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.1.5.5

Evaluate the exponent.

$\frac{5}{9} \cdot (\frac{58 - 2 \sqrt{301} - 29 \sqrt{301} + 301}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.2

Add $58$ and $301$ .

$\frac{5}{9} \cdot (\frac{359 - 2 \sqrt{301} - 29 \sqrt{301}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.10.3

Subtract $29 \sqrt{301}$ from $- 2 \sqrt{301}$ .

$\frac{5}{9} \cdot (\frac{359 - 31 \sqrt{301}}{36} ((\frac{5 - \sqrt{301}}{12}) - 3))$

Step 4.2.2.11

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{359 - 31 \sqrt{301}}{36} (\frac{5 - \sqrt{301}}{12} - 3 \cdot \frac{12}{12}))$

Step 4.2.2.12

Combine fractions.

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Step 4.2.2.12.1

Combine $- 3$ and $\frac{12}{12}$ .

$\frac{5}{9} \cdot (\frac{359 - 31 \sqrt{301}}{36} (\frac{5 - \sqrt{301}}{12} + \frac{- 3 \cdot 12}{12}))$

Step 4.2.2.12.2

Combine the numerators over the common denominator.

$\frac{5}{9} \cdot (\frac{359 - 31 \sqrt{301}}{36} \cdot \frac{5 - \sqrt{301} - 3 \cdot 12}{12})$

Step 4.2.2.13

Simplify the numerator.

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Step 4.2.2.13.1

Multiply $- 3$ by $12$ .

$\frac{5}{9} \cdot (\frac{359 - 31 \sqrt{301}}{36} \cdot \frac{5 - \sqrt{301} - 36}{12})$

Step 4.2.2.13.2

Subtract $36$ from $5$ .

$\frac{5}{9} \cdot (\frac{359 - 31 \sqrt{301}}{36} \cdot \frac{- 31 - \sqrt{301}}{12})$

Step 4.2.2.14

Multiply $\frac{359 - 31 \sqrt{301}}{36} \cdot \frac{- 31 - \sqrt{301}}{12}$ .

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Step 4.2.2.14.1

Multiply $\frac{359 - 31 \sqrt{301}}{36}$ by $\frac{- 31 - \sqrt{301}}{12}$ .

$\frac{5}{9} \cdot \frac{(359 - 31 \sqrt{301}) (- 31 - \sqrt{301})}{36 \cdot 12}$

Step 4.2.2.14.2

Multiply $36$ by $12$ .

$\frac{5}{9} \cdot \frac{(359 - 31 \sqrt{301}) (- 31 - \sqrt{301})}{432}$

Step 4.2.2.15

Expand $(359 - 31 \sqrt{301}) (- 31 - \sqrt{301})$ using the FOIL Method.

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Step 4.2.2.15.1

Apply the distributive property.

$\frac{5}{9} \cdot \frac{359 (- 31 - \sqrt{301}) - 31 \sqrt{301} (- 31 - \sqrt{301})}{432}$

Step 4.2.2.15.2

Apply the distributive property.

$\frac{5}{9} \cdot \frac{359 \cdot - 31 + 359 (- \sqrt{301}) - 31 \sqrt{301} (- 31 - \sqrt{301})}{432}$

Step 4.2.2.15.3

Apply the distributive property.

$\frac{5}{9} \cdot \frac{359 \cdot - 31 + 359 (- \sqrt{301}) - 31 \sqrt{301} \cdot - 31 - 31 \sqrt{301} (- \sqrt{301})}{432}$

Step 4.2.2.16

Simplify and combine like terms.

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Step 4.2.2.16.1

Simplify each term.

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Step 4.2.2.16.1.1

Multiply $359$ by $- 31$ .

$\frac{5}{9} \cdot \frac{- 11129 + 359 (- \sqrt{301}) - 31 \sqrt{301} \cdot - 31 - 31 \sqrt{301} (- \sqrt{301})}{432}$

Step 4.2.2.16.1.2

Multiply $- 1$ by $359$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} - 31 \sqrt{301} \cdot - 31 - 31 \sqrt{301} (- \sqrt{301})}{432}$

Step 4.2.2.16.1.3

Multiply $- 31$ by $- 31$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} - 31 \sqrt{301} (- \sqrt{301})}{432}$

Step 4.2.2.16.1.4

Multiply $- 31 \sqrt{301} (- \sqrt{301})$ .

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Step 4.2.2.16.1.4.1

Multiply $- 1$ by $- 31$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 \sqrt{301} \sqrt{301}}{432}$

Step 4.2.2.16.1.4.2

Raise $\sqrt{301}$ to the power of $1$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 ({\sqrt{301}}^{1} \sqrt{301})}{432}$

Step 4.2.2.16.1.4.3

Raise $\sqrt{301}$ to the power of $1$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 ({\sqrt{301}}^{1} {\sqrt{301}}^{1})}{432}$

Step 4.2.2.16.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 {\sqrt{301}}^{1 + 1}}{432}$

Step 4.2.2.16.1.4.5

Add $1$ and $1$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 {\sqrt{301}}^{2}}{432}$

Step 4.2.2.16.1.5

Rewrite ${\sqrt{301}}^{2}$ as $301$ .

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Step 4.2.2.16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{301}$ as $301^{\frac{1}{2}}$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 {(301^{\frac{1}{2}})}^{2}}{432}$

Step 4.2.2.16.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 \cdot 301^{\frac{1}{2} \cdot 2}}{432}$

Step 4.2.2.16.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 \cdot 301^{\frac{2}{2}}}{432}$

Step 4.2.2.16.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.16.1.5.4.1

Cancel the common factor.

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 \cdot 301^{\frac{2}{2}}}{432}$

Step 4.2.2.16.1.5.4.2

Rewrite the expression.

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 \cdot 301^{1}}{432}$

Step 4.2.2.16.1.5.5

Evaluate the exponent.

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 31 \cdot 301}{432}$

Step 4.2.2.16.1.6

Multiply $31$ by $301$ .

$\frac{5}{9} \cdot \frac{- 11129 - 359 \sqrt{301} + 961 \sqrt{301} + 9331}{432}$

Step 4.2.2.16.2

Add $- 11129$ and $9331$ .

$\frac{5}{9} \cdot \frac{- 1798 - 359 \sqrt{301} + 961 \sqrt{301}}{432}$

Step 4.2.2.16.3

Add $- 359 \sqrt{301}$ and $961 \sqrt{301}$ .

$\frac{5}{9} \cdot \frac{- 1798 + 602 \sqrt{301}}{432}$

Step 4.2.2.17

Cancel the common factor of $- 1798 + 602 \sqrt{301}$ and $432$ .

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Step 4.2.2.17.1

Factor $2$ out of $- 1798$ .

$\frac{5}{9} \cdot \frac{2 (- 899) + 602 \sqrt{301}}{432}$

Step 4.2.2.17.2

Factor $2$ out of $602 \sqrt{301}$ .

$\frac{5}{9} \cdot \frac{2 (- 899) + 2 (301 \sqrt{301})}{432}$

Step 4.2.2.17.3

Factor $2$ out of $2 (- 899) + 2 (301 \sqrt{301})$ .

$\frac{5}{9} \cdot \frac{2 (- 899 + 301 \sqrt{301})}{432}$

Step 4.2.2.17.4

Cancel the common factors.

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Step 4.2.2.17.4.1

Factor $2$ out of $432$ .

$\frac{5}{9} \cdot \frac{2 (- 899 + 301 \sqrt{301})}{2 \cdot 216}$

Step 4.2.2.17.4.2

Cancel the common factor.

$\frac{5}{9} \cdot \frac{2 (- 899 + 301 \sqrt{301})}{2 \cdot 216}$

Step 4.2.2.17.4.3

Rewrite the expression.

$\frac{5}{9} \cdot \frac{- 899 + 301 \sqrt{301}}{216}$

Step 4.2.2.18

Multiply $\frac{5}{9} \cdot \frac{- 899 + 301 \sqrt{301}}{216}$ .

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Step 4.2.2.18.1

Multiply $\frac{5}{9}$ by $\frac{- 899 + 301 \sqrt{301}}{216}$ .

$\frac{5 (- 899 + 301 \sqrt{301})}{9 \cdot 216}$

Step 4.2.2.18.2

Multiply $9$ by $216$ .

$\frac{5 (- 899 + 301 \sqrt{301})}{1944}$

Step 4.2.2.19

Rewrite $- 899$ as $- 1 (899)$ .

$\frac{5 (- 1 (899) + 301 \sqrt{301})}{1944}$

Step 4.2.2.20

Factor $- 1$ out of $301 \sqrt{301}$ .

$\frac{5 (- 1 (899) - (- 301 \sqrt{301}))}{1944}$

Step 4.2.2.21

Factor $- 1$ out of $- 1 (899) - (- 301 \sqrt{301})$ .

$\frac{5 (- 1 (899 - 301 \sqrt{301}))}{1944}$

Step 4.2.2.22

Move the negative in front of the fraction.

$- \frac{5 (899 - 301 \sqrt{301})}{1944}$

Step 4.3

List all of the points.

$(\frac{5 + \sqrt{301}}{12}, - \frac{5 (899 + 301 \sqrt{301})}{1944}), (\frac{5 - \sqrt{301}}{12}, - \frac{5 (899 - 301 \sqrt{301})}{1944})$

Step 5