Calculus Examples

$f (x) = 2 (x - 1) (x + 2) (3 x - 1)$

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

$2 \frac{d}{d x} [((x - 1) (x + 2)) (3 x - 1)]$

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 $f (x) = (x - 1) (x + 2)$ and $g (x) = 3 x - 1$ .

$2 ((x - 1) (x + 2) \frac{d}{d x} [3 x - 1] + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$2 ((x - 1) (x + 2) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$2 ((x - 1) (x + 2) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3.3

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

$2 ((x - 1) (x + 2) (3 \cdot 1 + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3.4

Multiply $3$ by $1$ .

$2 ((x - 1) (x + 2) (3 + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3.5

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

$2 ((x - 1) (x + 2) (3 + 0) + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3.6

Simplify the expression.

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Step 1.1.3.6.1

Add $3$ and $0$ .

$2 ((x - 1) (x + 2) \cdot 3 + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

Step 1.1.3.6.2

Move $3$ to the left of $(x - 1) (x + 2)$ .

$2 (3 \cdot ((x - 1) (x + 2)) + (3 x - 1) \frac{d}{d x} [(x - 1) (x + 2)])$

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 $f (x) = x - 1$ and $g (x) = x + 2$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) ((x - 1) \frac{d}{d x} [x + 2] + (x + 2) \frac{d}{d x} [x - 1]))$

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 $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) ((x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x - 1]))$

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 $n = 1$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) ((x - 1) (1 + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x - 1]))$

Step 1.1.5.3

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

$2 (3 (x - 1) (x + 2) + (3 x - 1) ((x - 1) (1 + 0) + (x + 2) \frac{d}{d x} [x - 1]))$

Step 1.1.5.4

Simplify the expression.

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Step 1.1.5.4.1

Add $1$ and $0$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) ((x - 1) \cdot 1 + (x + 2) \frac{d}{d x} [x - 1]))$

Step 1.1.5.4.2

Multiply $x - 1$ by $1$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) (x - 1 + (x + 2) \frac{d}{d x} [x - 1]))$

Step 1.1.5.5

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

$2 (3 (x - 1) (x + 2) + (3 x - 1) (x - 1 + (x + 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])))$

Step 1.1.5.6

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

$2 (3 (x - 1) (x + 2) + (3 x - 1) (x - 1 + (x + 2) (1 + \frac{d}{d x} [- 1])))$

Step 1.1.5.7

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

$2 (3 (x - 1) (x + 2) + (3 x - 1) (x - 1 + (x + 2) (1 + 0)))$

Step 1.1.5.8

Simplify by adding terms.

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Step 1.1.5.8.1

Add $1$ and $0$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) (x - 1 + (x + 2) \cdot 1))$

Step 1.1.5.8.2

Multiply $x + 2$ by $1$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) (x - 1 + x + 2))$

Step 1.1.5.8.3

Add $x$ and $x$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) (2 x - 1 + 2))$

Step 1.1.5.8.4

Add $- 1$ and $2$ .

$2 (3 (x - 1) (x + 2) + (3 x - 1) (2 x + 1))$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$2 ((3 x + 3 \cdot - 1) (x + 2) + (3 x - 1) (2 x + 1))$

Step 1.1.6.2

Apply the distributive property.

$2 ((3 x + 3 \cdot - 1) (x + 2)) + 2 ((3 x - 1) (2 x + 1))$

Step 1.1.6.3

Apply the distributive property.

$2 ((3 x + 3 \cdot - 1) x + (3 x + 3 \cdot - 1) \cdot 2) + 2 ((3 x - 1) (2 x + 1))$

Step 1.1.6.4

Apply the distributive property.

$2 (3 x \cdot x + 3 \cdot - 1 x + (3 x + 3 \cdot - 1) \cdot 2) + 2 ((3 x - 1) (2 x + 1))$

Step 1.1.6.5

Apply the distributive property.

$2 (3 x \cdot x + 3 \cdot - 1 x + 3 x \cdot 2 + 3 \cdot - 1 \cdot 2) + 2 ((3 x - 1) (2 x + 1))$

Step 1.1.6.6

Apply the distributive property.

$2 (3 x \cdot x) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 ((3 x - 1) (2 x + 1))$

Step 1.1.6.7

Apply the distributive property.

$2 (3 x \cdot x) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 ((3 x - 1) (2 x) + (3 x - 1) \cdot 1)$

Step 1.1.6.8

Apply the distributive property.

$2 (3 x \cdot x) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x) - 1 (2 x) + (3 x - 1) \cdot 1)$

Step 1.1.6.9

Apply the distributive property.

$2 (3 x \cdot x) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x) - 1 (2 x) + 3 x \cdot 1 - 1 \cdot 1)$

Step 1.1.6.10

Apply the distributive property.

$2 (3 x \cdot x) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11

Combine terms.

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Step 1.1.6.11.1

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

$2 (3 (x^{1} x)) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.2

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

$2 (3 (x^{1} x^{1})) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.3

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

$2 (3 x^{1 + 1}) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.4

Add $1$ and $1$ .

$2 (3 x^{2}) + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.5

Multiply $3$ by $2$ .

$6 x^{2} + 2 (3 \cdot - 1 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.6

Multiply $3$ by $- 1$ .

$6 x^{2} + 2 (- 3 x) + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.7

Multiply $- 3$ by $2$ .

$6 x^{2} - 6 x + 2 (3 x \cdot 2) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.8

Multiply $2$ by $3$ .

$6 x^{2} - 6 x + 2 (6 x) + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.9

Multiply $6$ by $2$ .

$6 x^{2} - 6 x + 12 x + 2 (3 \cdot - 1 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.10

Multiply $3$ by $- 1$ .

$6 x^{2} - 6 x + 12 x + 2 (- 3 \cdot 2) + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.11

Multiply $- 3$ by $2$ .

$6 x^{2} - 6 x + 12 x + 2 \cdot - 6 + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.12

Multiply $2$ by $- 6$ .

$6 x^{2} - 6 x + 12 x - 12 + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.13

Add $- 6 x$ and $12 x$ .

$6 x^{2} + 6 x - 12 + 2 (3 x (2 x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.14

Multiply $2$ by $3$ .

$6 x^{2} + 6 x - 12 + 2 (6 x \cdot x) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.15

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

$6 x^{2} + 6 x - 12 + 2 (6 (x^{1} x)) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.16

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

$6 x^{2} + 6 x - 12 + 2 (6 (x^{1} x^{1})) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.17

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

$6 x^{2} + 6 x - 12 + 2 (6 x^{1 + 1}) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.18

Add $1$ and $1$ .

$6 x^{2} + 6 x - 12 + 2 (6 x^{2}) + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.19

Multiply $6$ by $2$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} + 2 (- 1 (2 x)) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.20

Multiply $2$ by $- 1$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} + 2 (- 2 x) + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.21

Multiply $- 2$ by $2$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} - 4 x + 2 (3 x \cdot 1) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.22

Multiply $3$ by $1$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} - 4 x + 2 (3 x) + 2 (- 1 \cdot 1)$

Step 1.1.6.11.23

Multiply $3$ by $2$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} - 4 x + 6 x + 2 (- 1 \cdot 1)$

Step 1.1.6.11.24

Multiply $- 1$ by $1$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} - 4 x + 6 x + 2 \cdot - 1$

Step 1.1.6.11.25

Multiply $2$ by $- 1$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} - 4 x + 6 x - 2$

Step 1.1.6.11.26

Add $- 4 x$ and $6 x$ .

$6 x^{2} + 6 x - 12 + 12 x^{2} + 2 x - 2$

Step 1.1.6.11.27

Add $6 x^{2}$ and $12 x^{2}$ .

$18 x^{2} + 6 x - 12 + 2 x - 2$

Step 1.1.6.11.28

Add $6 x$ and $2 x$ .

$18 x^{2} + 8 x - 12 - 2$

Step 1.1.6.11.29

Subtract $2$ from $- 12$ .

$f' (x) = 18 x^{2} + 8 x - 14$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $18 x^{2} + 8 x - 14$ .

$18 x^{2} + 8 x - 14$

Step 2

Set the first derivative equal to $0$ then solve the equation $18 x^{2} + 8 x - 14 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$18 x^{2} + 8 x - 14 = 0$

Step 2.2

Factor $2$ out of $18 x^{2} + 8 x - 14$ .

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Step 2.2.1

Factor $2$ out of $18 x^{2}$ .

$2 (9 x^{2}) + 8 x - 14 = 0$

Step 2.2.2

Factor $2$ out of $8 x$ .

$2 (9 x^{2}) + 2 (4 x) - 14 = 0$

Step 2.2.3

Factor $2$ out of $- 14$ .

$2 (9 x^{2}) + 2 (4 x) + 2 (- 7) = 0$

Step 2.2.4

Factor $2$ out of $2 (9 x^{2}) + 2 (4 x)$ .

$2 (9 x^{2} + 4 x) + 2 (- 7) = 0$

Step 2.2.5

Factor $2$ out of $2 (9 x^{2} + 4 x) + 2 (- 7)$ .

$2 (9 x^{2} + 4 x - 7) = 0$

Step 2.3

Divide each term in $2 (9 x^{2} + 4 x - 7) = 0$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 (9 x^{2} + 4 x - 7) = 0$ by $2$ .

$\frac{2 (9 x^{2} + 4 x - 7)}{2} = \frac{0}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 (9 x^{2} + 4 x - 7)}{2} = \frac{0}{2}$

Step 2.3.2.1.2

Divide $9 x^{2} + 4 x - 7$ by $1$ .

$9 x^{2} + 4 x - 7 = \frac{0}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $2$ .

$9 x^{2} + 4 x - 7 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 9$ , $b = 4$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (9 \cdot - 7)}}{2 \cdot 9}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 9 \cdot - 7}}{2 \cdot 9}$

Step 2.6.1.2

Multiply $- 4 \cdot 9 \cdot - 7$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 4 \pm \sqrt{16 - 36 \cdot - 7}}{2 \cdot 9}$

Step 2.6.1.2.2

Multiply $- 36$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 252}}{2 \cdot 9}$

Step 2.6.1.3

Add $16$ and $252$ .

$x = \frac{- 4 \pm \sqrt{268}}{2 \cdot 9}$

Step 2.6.1.4

Rewrite $268$ as $2^{2} \cdot 67$ .

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Step 2.6.1.4.1

Factor $4$ out of $268$ .

$x = \frac{- 4 \pm \sqrt{4 (67)}}{2 \cdot 9}$

Step 2.6.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 67}}{2 \cdot 9}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{67}}{2 \cdot 9}$

Step 2.6.2

Multiply $2$ by $9$ .

$x = \frac{- 4 \pm 2 \sqrt{67}}{18}$

Step 2.6.3

Simplify $\frac{- 4 \pm 2 \sqrt{67}}{18}$ .

$x = \frac{- 2 \pm \sqrt{67}}{9}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 9 \cdot - 7}}{2 \cdot 9}$

Step 2.7.1.2

Multiply $- 4 \cdot 9 \cdot - 7$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 4 \pm \sqrt{16 - 36 \cdot - 7}}{2 \cdot 9}$

Step 2.7.1.2.2

Multiply $- 36$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 252}}{2 \cdot 9}$

Step 2.7.1.3

Add $16$ and $252$ .

$x = \frac{- 4 \pm \sqrt{268}}{2 \cdot 9}$

Step 2.7.1.4

Rewrite $268$ as $2^{2} \cdot 67$ .

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Step 2.7.1.4.1

Factor $4$ out of $268$ .

$x = \frac{- 4 \pm \sqrt{4 (67)}}{2 \cdot 9}$

Step 2.7.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 67}}{2 \cdot 9}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{67}}{2 \cdot 9}$

Step 2.7.2

Multiply $2$ by $9$ .

$x = \frac{- 4 \pm 2 \sqrt{67}}{18}$

Step 2.7.3

Simplify $\frac{- 4 \pm 2 \sqrt{67}}{18}$ .

$x = \frac{- 2 \pm \sqrt{67}}{9}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + \sqrt{67}}{9}$

Step 2.7.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + \sqrt{67}}{9}$

Step 2.7.6

Factor $- 1$ out of $\sqrt{67}$ .

$x = \frac{- 1 \cdot 2 - 1 (- \sqrt{67})}{9}$

Step 2.7.7

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{67})$ .

$x = \frac{- 1 (2 - \sqrt{67})}{9}$

Step 2.7.8

Move the negative in front of the fraction.

$x = - \frac{2 - \sqrt{67}}{9}$

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 $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 9 \cdot - 7}}{2 \cdot 9}$

Step 2.8.1.2

Multiply $- 4 \cdot 9 \cdot - 7$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 4 \pm \sqrt{16 - 36 \cdot - 7}}{2 \cdot 9}$

Step 2.8.1.2.2

Multiply $- 36$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 252}}{2 \cdot 9}$

Step 2.8.1.3

Add $16$ and $252$ .

$x = \frac{- 4 \pm \sqrt{268}}{2 \cdot 9}$

Step 2.8.1.4

Rewrite $268$ as $2^{2} \cdot 67$ .

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Step 2.8.1.4.1

Factor $4$ out of $268$ .

$x = \frac{- 4 \pm \sqrt{4 (67)}}{2 \cdot 9}$

Step 2.8.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 67}}{2 \cdot 9}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{67}}{2 \cdot 9}$

Step 2.8.2

Multiply $2$ by $9$ .

$x = \frac{- 4 \pm 2 \sqrt{67}}{18}$

Step 2.8.3

Simplify $\frac{- 4 \pm 2 \sqrt{67}}{18}$ .

$x = \frac{- 2 \pm \sqrt{67}}{9}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - \sqrt{67}}{9}$

Step 2.8.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - \sqrt{67}}{9}$

Step 2.8.6

Factor $- 1$ out of $- \sqrt{67}$ .

$x = \frac{- 1 \cdot 2 - (\sqrt{67})}{9}$

Step 2.8.7

Factor $- 1$ out of $- 1 (2) - (\sqrt{67})$ .

$x = \frac{- 1 (2 + \sqrt{67})}{9}$

Step 2.8.8

Move the negative in front of the fraction.

$x = - \frac{2 + \sqrt{67}}{9}$

Step 2.9

The final answer is the combination of both solutions.

$x = - \frac{2 - \sqrt{67}}{9}, - \frac{2 + \sqrt{67}}{9}$

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 $2 (x - 1) (x + 2) (3 x - 1)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{2 - \sqrt{67}}{9}$ .

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Step 4.1.1

Substitute $- \frac{2 - \sqrt{67}}{9}$ for $x$ .

$2 ((- \frac{2 - \sqrt{67}}{9}) - 1) ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$2 (- \frac{2 - \sqrt{67}}{9} - 1 \cdot \frac{9}{9}) ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.2

Combine fractions.

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Step 4.1.2.2.1

Combine $- 1$ and $\frac{9}{9}$ .

$2 (- \frac{2 - \sqrt{67}}{9} + \frac{- 1 \cdot 9}{9}) ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$2 \frac{- (2 - \sqrt{67}) - 1 \cdot 9}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Apply the distributive property.

$2 \frac{- 1 \cdot 2 - - \sqrt{67} - 1 \cdot 9}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.3.2

Multiply $- 1$ by $2$ .

$2 \frac{- 2 - - \sqrt{67} - 1 \cdot 9}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.3.3

Multiply $- - \sqrt{67}$ .

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Step 4.1.2.3.3.1

Multiply $- 1$ by $- 1$ .

$2 \frac{- 2 + 1 \sqrt{67} - 1 \cdot 9}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.3.3.2

Multiply $\sqrt{67}$ by $1$ .

$2 \frac{- 2 + \sqrt{67} - 1 \cdot 9}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.3.4

Multiply $- 1$ by $9$ .

$2 \frac{- 2 + \sqrt{67} - 9}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.3.5

Subtract $9$ from $- 2$ .

$2 \frac{- 11 + \sqrt{67}}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.4

Combine $2$ and $\frac{- 11 + \sqrt{67}}{9}$ .

$\frac{2 (- 11 + \sqrt{67})}{9} ((- \frac{2 - \sqrt{67}}{9}) + 2) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{2 (- 11 + \sqrt{67})}{9} (- \frac{2 - \sqrt{67}}{9} + 2 \cdot \frac{9}{9}) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.6

Combine fractions.

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Step 4.1.2.6.1

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

$\frac{2 (- 11 + \sqrt{67})}{9} (- \frac{2 - \sqrt{67}}{9} + \frac{2 \cdot 9}{9}) (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.6.2

Combine the numerators over the common denominator.

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{- (2 - \sqrt{67}) + 2 \cdot 9}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.7

Simplify the numerator.

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Step 4.1.2.7.1

Apply the distributive property.

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{- 1 \cdot 2 - - \sqrt{67} + 2 \cdot 9}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.7.2

Multiply $- 1$ by $2$ .

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{- 2 - - \sqrt{67} + 2 \cdot 9}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.7.3

Multiply $- - \sqrt{67}$ .

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Step 4.1.2.7.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{- 2 + 1 \sqrt{67} + 2 \cdot 9}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.7.3.2

Multiply $\sqrt{67}$ by $1$ .

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{- 2 + \sqrt{67} + 2 \cdot 9}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.7.4

Multiply $2$ by $9$ .

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{- 2 + \sqrt{67} + 18}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.7.5

Add $- 2$ and $18$ .

$\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{16 + \sqrt{67}}{9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.8

Multiply $\frac{2 (- 11 + \sqrt{67})}{9} \cdot \frac{16 + \sqrt{67}}{9}$ .

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Step 4.1.2.8.1

Multiply $\frac{2 (- 11 + \sqrt{67})}{9}$ by $\frac{16 + \sqrt{67}}{9}$ .

$\frac{2 (- 11 + \sqrt{67}) (16 + \sqrt{67})}{9 \cdot 9} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.8.2

Multiply $9$ by $9$ .

$\frac{2 (- 11 + \sqrt{67}) (16 + \sqrt{67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.9

Group $16 + \sqrt{67}$ and $- 11 + \sqrt{67}$ together.

$\frac{2 ((16 + \sqrt{67}) (- 11 + \sqrt{67}))}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.10

Expand $(16 + \sqrt{67}) (- 11 + \sqrt{67})$ using the FOIL Method.

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Step 4.1.2.10.1

Apply the distributive property.

$\frac{2 (16 (- 11 + \sqrt{67}) + \sqrt{67} (- 11 + \sqrt{67}))}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.10.2

Apply the distributive property.

$\frac{2 (16 \cdot - 11 + 16 \sqrt{67} + \sqrt{67} (- 11 + \sqrt{67}))}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.10.3

Apply the distributive property.

$\frac{2 (16 \cdot - 11 + 16 \sqrt{67} + \sqrt{67} \cdot - 11 + \sqrt{67} \sqrt{67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11

Simplify and combine like terms.

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Step 4.1.2.11.1

Simplify each term.

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Step 4.1.2.11.1.1

Multiply $16$ by $- 11$ .

$\frac{2 (- 176 + 16 \sqrt{67} + \sqrt{67} \cdot - 11 + \sqrt{67} \sqrt{67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.1.2

Move $- 11$ to the left of $\sqrt{67}$ .

$\frac{2 (- 176 + 16 \sqrt{67} - 11 \cdot \sqrt{67} + \sqrt{67} \sqrt{67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.1.3

Combine using the product rule for radicals.

$\frac{2 (- 176 + 16 \sqrt{67} - 11 \sqrt{67} + \sqrt{67 \cdot 67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.1.4

Multiply $67$ by $67$ .

$\frac{2 (- 176 + 16 \sqrt{67} - 11 \sqrt{67} + \sqrt{4489})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.1.5

Rewrite $4489$ as $67^{2}$ .

$\frac{2 (- 176 + 16 \sqrt{67} - 11 \sqrt{67} + \sqrt{67^{2}})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.1.6

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

$\frac{2 (- 176 + 16 \sqrt{67} - 11 \sqrt{67} + 67)}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.2

Add $- 176$ and $67$ .

$\frac{2 (- 109 + 16 \sqrt{67} - 11 \sqrt{67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.11.3

Subtract $11 \sqrt{67}$ from $16 \sqrt{67}$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (3 (- \frac{2 - \sqrt{67}}{9}) - 1)$

Step 4.1.2.12

Simplify each term.

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Step 4.1.2.12.1

Cancel the common factor of $3$ .

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Step 4.1.2.12.1.1

Move the leading negative in $- \frac{2 - \sqrt{67}}{9}$ into the numerator.

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (3 \frac{- (2 - \sqrt{67})}{9} - 1)$

Step 4.1.2.12.1.2

Factor $3$ out of $9$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (3 \frac{- (2 - \sqrt{67})}{3 (3)} - 1)$

Step 4.1.2.12.1.3

Cancel the common factor.

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (3 \frac{- (2 - \sqrt{67})}{3 \cdot 3} - 1)$

Step 4.1.2.12.1.4

Rewrite the expression.

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (\frac{- (2 - \sqrt{67})}{3} - 1)$

Step 4.1.2.12.2

Move the negative in front of the fraction.

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (- \frac{2 - \sqrt{67}}{3} - 1)$

Step 4.1.2.13

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

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (- \frac{2 - \sqrt{67}}{3} - 1 \cdot \frac{3}{3})$

Step 4.1.2.14

Combine fractions.

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Step 4.1.2.14.1

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

$\frac{2 (- 109 + 5 \sqrt{67})}{81} (- \frac{2 - \sqrt{67}}{3} + \frac{- 1 \cdot 3}{3})$

Step 4.1.2.14.2

Combine the numerators over the common denominator.

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- (2 - \sqrt{67}) - 1 \cdot 3}{3}$

Step 4.1.2.15

Simplify the numerator.

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Step 4.1.2.15.1

Apply the distributive property.

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 1 \cdot 2 - - \sqrt{67} - 1 \cdot 3}{3}$

Step 4.1.2.15.2

Multiply $- 1$ by $2$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 2 - - \sqrt{67} - 1 \cdot 3}{3}$

Step 4.1.2.15.3

Multiply $- - \sqrt{67}$ .

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Step 4.1.2.15.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 2 + 1 \sqrt{67} - 1 \cdot 3}{3}$

Step 4.1.2.15.3.2

Multiply $\sqrt{67}$ by $1$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 2 + \sqrt{67} - 1 \cdot 3}{3}$

Step 4.1.2.15.4

Multiply $- 1$ by $3$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 2 + \sqrt{67} - 3}{3}$

Step 4.1.2.15.5

Subtract $3$ from $- 2$ .

$\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 5 + \sqrt{67}}{3}$

Step 4.1.2.16

Multiply $\frac{2 (- 109 + 5 \sqrt{67})}{81} \cdot \frac{- 5 + \sqrt{67}}{3}$ .

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Step 4.1.2.16.1

Multiply $\frac{2 (- 109 + 5 \sqrt{67})}{81}$ by $\frac{- 5 + \sqrt{67}}{3}$ .

$\frac{2 (- 109 + 5 \sqrt{67}) (- 5 + \sqrt{67})}{81 \cdot 3}$

Step 4.1.2.16.2

Multiply $81$ by $3$ .

$\frac{2 (- 109 + 5 \sqrt{67}) (- 5 + \sqrt{67})}{243}$

Step 4.1.2.17

Group $- 5 + \sqrt{67}$ and $- 109 + 5 \sqrt{67}$ together.

$\frac{2 ((- 5 + \sqrt{67}) (- 109 + 5 \sqrt{67}))}{243}$

Step 4.1.2.18

Expand $(- 5 + \sqrt{67}) (- 109 + 5 \sqrt{67})$ using the FOIL Method.

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Step 4.1.2.18.1

Apply the distributive property.

$\frac{2 (- 5 (- 109 + 5 \sqrt{67}) + \sqrt{67} (- 109 + 5 \sqrt{67}))}{243}$

Step 4.1.2.18.2

Apply the distributive property.

$\frac{2 (- 5 \cdot - 109 - 5 (5 \sqrt{67}) + \sqrt{67} (- 109 + 5 \sqrt{67}))}{243}$

Step 4.1.2.18.3

Apply the distributive property.

$\frac{2 (- 5 \cdot - 109 - 5 (5 \sqrt{67}) + \sqrt{67} \cdot - 109 + \sqrt{67} (5 \sqrt{67}))}{243}$

Step 4.1.2.19

Simplify and combine like terms.

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Step 4.1.2.19.1

Simplify each term.

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Step 4.1.2.19.1.1

Multiply $- 5$ by $- 109$ .

$\frac{2 (545 - 5 (5 \sqrt{67}) + \sqrt{67} \cdot - 109 + \sqrt{67} (5 \sqrt{67}))}{243}$

Step 4.1.2.19.1.2

Multiply $5$ by $- 5$ .

$\frac{2 (545 - 25 \sqrt{67} + \sqrt{67} \cdot - 109 + \sqrt{67} (5 \sqrt{67}))}{243}$

Step 4.1.2.19.1.3

Move $- 109$ to the left of $\sqrt{67}$ .

$\frac{2 (545 - 25 \sqrt{67} - 109 \cdot \sqrt{67} + \sqrt{67} (5 \sqrt{67}))}{243}$

Step 4.1.2.19.1.4

Multiply $\sqrt{67} (5 \sqrt{67})$ .

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Step 4.1.2.19.1.4.1

Raise $\sqrt{67}$ to the power of $1$ .

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 ({\sqrt{67}}^{1} \sqrt{67}))}{243}$

Step 4.1.2.19.1.4.2

Raise $\sqrt{67}$ to the power of $1$ .

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 ({\sqrt{67}}^{1} {\sqrt{67}}^{1}))}{243}$

Step 4.1.2.19.1.4.3

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

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 {\sqrt{67}}^{1 + 1})}{243}$

Step 4.1.2.19.1.4.4

Add $1$ and $1$ .

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 {\sqrt{67}}^{2})}{243}$

Step 4.1.2.19.1.5

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 4.1.2.19.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 {(67^{\frac{1}{2}})}^{2})}{243}$

Step 4.1.2.19.1.5.2

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

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 \cdot 67^{\frac{1}{2} \cdot 2})}{243}$

Step 4.1.2.19.1.5.3

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

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 \cdot 67^{\frac{2}{2}})}{243}$

Step 4.1.2.19.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.2.19.1.5.4.1

Cancel the common factor.

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 \cdot 67^{\frac{2}{2}})}{243}$

Step 4.1.2.19.1.5.4.2

Rewrite the expression.

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 \cdot 67^{1})}{243}$

Step 4.1.2.19.1.5.5

Evaluate the exponent.

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 5 \cdot 67)}{243}$

Step 4.1.2.19.1.6

Multiply $5$ by $67$ .

$\frac{2 (545 - 25 \sqrt{67} - 109 \sqrt{67} + 335)}{243}$

Step 4.1.2.19.2

Add $545$ and $335$ .

$\frac{2 (880 - 25 \sqrt{67} - 109 \sqrt{67})}{243}$

Step 4.1.2.19.3

Subtract $109 \sqrt{67}$ from $- 25 \sqrt{67}$ .

$\frac{2 (880 - 134 \sqrt{67})}{243}$

Step 4.2

Evaluate at $x = - \frac{2 + \sqrt{67}}{9}$ .

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Step 4.2.1

Substitute $- \frac{2 + \sqrt{67}}{9}$ for $x$ .

$2 ((- \frac{2 + \sqrt{67}}{9}) - 1) ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$2 (- \frac{2 + \sqrt{67}}{9} - 1 \cdot \frac{9}{9}) ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.2

Combine fractions.

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Step 4.2.2.2.1

Combine $- 1$ and $\frac{9}{9}$ .

$2 (- \frac{2 + \sqrt{67}}{9} + \frac{- 1 \cdot 9}{9}) ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.2.2

Combine the numerators over the common denominator.

$2 \frac{- (2 + \sqrt{67}) - 1 \cdot 9}{9} ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.3

Simplify the numerator.

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Step 4.2.2.3.1

Apply the distributive property.

$2 \frac{- 1 \cdot 2 - \sqrt{67} - 1 \cdot 9}{9} ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.3.2

Multiply $- 1$ by $2$ .

$2 \frac{- 2 - \sqrt{67} - 1 \cdot 9}{9} ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.3.3

Multiply $- 1$ by $9$ .

$2 \frac{- 2 - \sqrt{67} - 9}{9} ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.3.4

Subtract $9$ from $- 2$ .

$2 \frac{- 11 - \sqrt{67}}{9} ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.4

Combine $2$ and $\frac{- 11 - \sqrt{67}}{9}$ .

$\frac{2 (- 11 - \sqrt{67})}{9} ((- \frac{2 + \sqrt{67}}{9}) + 2) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{2 (- 11 - \sqrt{67})}{9} (- \frac{2 + \sqrt{67}}{9} + 2 \cdot \frac{9}{9}) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.6

Combine fractions.

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Step 4.2.2.6.1

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

$\frac{2 (- 11 - \sqrt{67})}{9} (- \frac{2 + \sqrt{67}}{9} + \frac{2 \cdot 9}{9}) (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.6.2

Combine the numerators over the common denominator.

$\frac{2 (- 11 - \sqrt{67})}{9} \cdot \frac{- (2 + \sqrt{67}) + 2 \cdot 9}{9} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Apply the distributive property.

$\frac{2 (- 11 - \sqrt{67})}{9} \cdot \frac{- 1 \cdot 2 - \sqrt{67} + 2 \cdot 9}{9} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.7.2

Multiply $- 1$ by $2$ .

$\frac{2 (- 11 - \sqrt{67})}{9} \cdot \frac{- 2 - \sqrt{67} + 2 \cdot 9}{9} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.7.3

Multiply $2$ by $9$ .

$\frac{2 (- 11 - \sqrt{67})}{9} \cdot \frac{- 2 - \sqrt{67} + 18}{9} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.7.4

Add $- 2$ and $18$ .

$\frac{2 (- 11 - \sqrt{67})}{9} \cdot \frac{16 - \sqrt{67}}{9} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.8

Multiply $\frac{2 (- 11 - \sqrt{67})}{9} \cdot \frac{16 - \sqrt{67}}{9}$ .

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Step 4.2.2.8.1

Multiply $\frac{2 (- 11 - \sqrt{67})}{9}$ by $\frac{16 - \sqrt{67}}{9}$ .

$\frac{2 (- 11 - \sqrt{67}) (16 - \sqrt{67})}{9 \cdot 9} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.8.2

Multiply $9$ by $9$ .

$\frac{2 (- 11 - \sqrt{67}) (16 - \sqrt{67})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.9

Group $16 - \sqrt{67}$ and $- 11 - \sqrt{67}$ together.

$\frac{2 ((16 - \sqrt{67}) (- 11 - \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.10

Expand $(16 - \sqrt{67}) (- 11 - \sqrt{67})$ using the FOIL Method.

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Step 4.2.2.10.1

Apply the distributive property.

$\frac{2 (16 (- 11 - \sqrt{67}) - \sqrt{67} (- 11 - \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.10.2

Apply the distributive property.

$\frac{2 (16 \cdot - 11 + 16 (- \sqrt{67}) - \sqrt{67} (- 11 - \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.10.3

Apply the distributive property.

$\frac{2 (16 \cdot - 11 + 16 (- \sqrt{67}) - \sqrt{67} \cdot - 11 - \sqrt{67} (- \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11

Simplify and combine like terms.

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Step 4.2.2.11.1

Simplify each term.

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Step 4.2.2.11.1.1

Multiply $16$ by $- 11$ .

$\frac{2 (- 176 + 16 (- \sqrt{67}) - \sqrt{67} \cdot - 11 - \sqrt{67} (- \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.2

Multiply $- 1$ by $16$ .

$\frac{2 (- 176 - 16 \sqrt{67} - \sqrt{67} \cdot - 11 - \sqrt{67} (- \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.3

Multiply $- 11$ by $- 1$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} - \sqrt{67} (- \sqrt{67}))}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.4

Multiply $- \sqrt{67} (- \sqrt{67})$ .

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Step 4.2.2.11.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + 1 \sqrt{67} \sqrt{67})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.4.2

Multiply $\sqrt{67}$ by $1$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + \sqrt{67} \sqrt{67})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.4.3

Raise $\sqrt{67}$ to the power of $1$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + {\sqrt{67}}^{1} \sqrt{67})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.4.4

Raise $\sqrt{67}$ to the power of $1$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + {\sqrt{67}}^{1} {\sqrt{67}}^{1})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.4.5

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

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + {\sqrt{67}}^{1 + 1})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.4.6

Add $1$ and $1$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + {\sqrt{67}}^{2})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.5

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 4.2.2.11.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + {(67^{\frac{1}{2}})}^{2})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.5.2

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

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + 67^{\frac{1}{2} \cdot 2})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.5.3

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

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + 67^{\frac{2}{2}})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.11.1.5.4.1

Cancel the common factor.

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + 67^{\frac{2}{2}})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.5.4.2

Rewrite the expression.

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + 67^{1})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.1.5.5

Evaluate the exponent.

$\frac{2 (- 176 - 16 \sqrt{67} + 11 \sqrt{67} + 67)}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.2

Add $- 176$ and $67$ .

$\frac{2 (- 109 - 16 \sqrt{67} + 11 \sqrt{67})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.11.3

Add $- 16 \sqrt{67}$ and $11 \sqrt{67}$ .

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (3 (- \frac{2 + \sqrt{67}}{9}) - 1)$

Step 4.2.2.12

Simplify each term.

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Step 4.2.2.12.1

Cancel the common factor of $3$ .

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Step 4.2.2.12.1.1

Move the leading negative in $- \frac{2 + \sqrt{67}}{9}$ into the numerator.

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (3 \frac{- (2 + \sqrt{67})}{9} - 1)$

Step 4.2.2.12.1.2

Factor $3$ out of $9$ .

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (3 \frac{- (2 + \sqrt{67})}{3 (3)} - 1)$

Step 4.2.2.12.1.3

Cancel the common factor.

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (3 \frac{- (2 + \sqrt{67})}{3 \cdot 3} - 1)$

Step 4.2.2.12.1.4

Rewrite the expression.

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (\frac{- (2 + \sqrt{67})}{3} - 1)$

Step 4.2.2.12.2

Move the negative in front of the fraction.

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (- \frac{2 + \sqrt{67}}{3} - 1)$

Step 4.2.2.13

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

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (- \frac{2 + \sqrt{67}}{3} - 1 \cdot \frac{3}{3})$

Step 4.2.2.14

Combine fractions.

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Step 4.2.2.14.1

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

$\frac{2 (- 109 - 5 \sqrt{67})}{81} (- \frac{2 + \sqrt{67}}{3} + \frac{- 1 \cdot 3}{3})$

Step 4.2.2.14.2

Combine the numerators over the common denominator.

$\frac{2 (- 109 - 5 \sqrt{67})}{81} \cdot \frac{- (2 + \sqrt{67}) - 1 \cdot 3}{3}$

Step 4.2.2.15

Simplify the numerator.

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Step 4.2.2.15.1

Apply the distributive property.

$\frac{2 (- 109 - 5 \sqrt{67})}{81} \cdot \frac{- 1 \cdot 2 - \sqrt{67} - 1 \cdot 3}{3}$

Step 4.2.2.15.2

Multiply $- 1$ by $2$ .

$\frac{2 (- 109 - 5 \sqrt{67})}{81} \cdot \frac{- 2 - \sqrt{67} - 1 \cdot 3}{3}$

Step 4.2.2.15.3

Multiply $- 1$ by $3$ .

$\frac{2 (- 109 - 5 \sqrt{67})}{81} \cdot \frac{- 2 - \sqrt{67} - 3}{3}$

Step 4.2.2.15.4

Subtract $3$ from $- 2$ .

$\frac{2 (- 109 - 5 \sqrt{67})}{81} \cdot \frac{- 5 - \sqrt{67}}{3}$

Step 4.2.2.16

Multiply $\frac{2 (- 109 - 5 \sqrt{67})}{81} \cdot \frac{- 5 - \sqrt{67}}{3}$ .

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Step 4.2.2.16.1

Multiply $\frac{2 (- 109 - 5 \sqrt{67})}{81}$ by $\frac{- 5 - \sqrt{67}}{3}$ .

$\frac{2 (- 109 - 5 \sqrt{67}) (- 5 - \sqrt{67})}{81 \cdot 3}$

Step 4.2.2.16.2

Multiply $81$ by $3$ .

$\frac{2 (- 109 - 5 \sqrt{67}) (- 5 - \sqrt{67})}{243}$

Step 4.2.2.17

Group $- 5 - \sqrt{67}$ and $- 109 - 5 \sqrt{67}$ together.

$\frac{2 ((- 5 - \sqrt{67}) (- 109 - 5 \sqrt{67}))}{243}$

Step 4.2.2.18

Expand $(- 5 - \sqrt{67}) (- 109 - 5 \sqrt{67})$ using the FOIL Method.

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Step 4.2.2.18.1

Apply the distributive property.

$\frac{2 (- 5 (- 109 - 5 \sqrt{67}) - \sqrt{67} (- 109 - 5 \sqrt{67}))}{243}$

Step 4.2.2.18.2

Apply the distributive property.

$\frac{2 (- 5 \cdot - 109 - 5 (- 5 \sqrt{67}) - \sqrt{67} (- 109 - 5 \sqrt{67}))}{243}$

Step 4.2.2.18.3

Apply the distributive property.

$\frac{2 (- 5 \cdot - 109 - 5 (- 5 \sqrt{67}) - \sqrt{67} \cdot - 109 - \sqrt{67} (- 5 \sqrt{67}))}{243}$

Step 4.2.2.19

Simplify and combine like terms.

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Step 4.2.2.19.1

Simplify each term.

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Step 4.2.2.19.1.1

Multiply $- 5$ by $- 109$ .

$\frac{2 (545 - 5 (- 5 \sqrt{67}) - \sqrt{67} \cdot - 109 - \sqrt{67} (- 5 \sqrt{67}))}{243}$

Step 4.2.2.19.1.2

Multiply $- 5$ by $- 5$ .

$\frac{2 (545 + 25 \sqrt{67} - \sqrt{67} \cdot - 109 - \sqrt{67} (- 5 \sqrt{67}))}{243}$

Step 4.2.2.19.1.3

Multiply $- 109$ by $- 1$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} - \sqrt{67} (- 5 \sqrt{67}))}{243}$

Step 4.2.2.19.1.4

Multiply $- \sqrt{67} (- 5 \sqrt{67})$ .

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Step 4.2.2.19.1.4.1

Multiply $- 5$ by $- 1$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 \sqrt{67} \sqrt{67})}{243}$

Step 4.2.2.19.1.4.2

Raise $\sqrt{67}$ to the power of $1$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 ({\sqrt{67}}^{1} \sqrt{67}))}{243}$

Step 4.2.2.19.1.4.3

Raise $\sqrt{67}$ to the power of $1$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 ({\sqrt{67}}^{1} {\sqrt{67}}^{1}))}{243}$

Step 4.2.2.19.1.4.4

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

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 {\sqrt{67}}^{1 + 1})}{243}$

Step 4.2.2.19.1.4.5

Add $1$ and $1$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 {\sqrt{67}}^{2})}{243}$

Step 4.2.2.19.1.5

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 4.2.2.19.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 {(67^{\frac{1}{2}})}^{2})}{243}$

Step 4.2.2.19.1.5.2

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

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 \cdot 67^{\frac{1}{2} \cdot 2})}{243}$

Step 4.2.2.19.1.5.3

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

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 \cdot 67^{\frac{2}{2}})}{243}$

Step 4.2.2.19.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.19.1.5.4.1

Cancel the common factor.

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 \cdot 67^{\frac{2}{2}})}{243}$

Step 4.2.2.19.1.5.4.2

Rewrite the expression.

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 \cdot 67^{1})}{243}$

Step 4.2.2.19.1.5.5

Evaluate the exponent.

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 5 \cdot 67)}{243}$

Step 4.2.2.19.1.6

Multiply $5$ by $67$ .

$\frac{2 (545 + 25 \sqrt{67} + 109 \sqrt{67} + 335)}{243}$

Step 4.2.2.19.2

Add $545$ and $335$ .

$\frac{2 (880 + 25 \sqrt{67} + 109 \sqrt{67})}{243}$

Step 4.2.2.19.3

Add $25 \sqrt{67}$ and $109 \sqrt{67}$ .

$\frac{2 (880 + 134 \sqrt{67})}{243}$

Step 4.3

List all of the points.

$(- \frac{2 - \sqrt{67}}{9}, \frac{2 (880 - 134 \sqrt{67})}{243}), (- \frac{2 + \sqrt{67}}{9}, \frac{2 (880 + 134 \sqrt{67})}{243})$

Step 5