Calculus Examples

$5 x^{2} + 4 y^{2} - 10 x + 24 y + 8 = 0$

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 4$ , $b = 24$ , and $c = 5 x^{2} - 10 x + 8$ into the quadratic formula and solve for $y$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (4 \cdot (5 x^{2} - 10 x + 8))}}{2 \cdot 4}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 4 \cdot (5 x^{2} - 10 x + 8)}}{2 \cdot 4}$

Step 1.3.1.2

Multiply $- 4$ by $4$ .

$y = \frac{- 24 \pm \sqrt{576 - 16 \cdot (5 x^{2} - 10 x + 8)}}{2 \cdot 4}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{576 - 16 (5 x^{2}) - 16 (- 10 x) - 16 \cdot 8}}{2 \cdot 4}$

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $5$ by $- 16$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} - 16 (- 10 x) - 16 \cdot 8}}{2 \cdot 4}$

Step 1.3.1.4.2

Multiply $- 10$ by $- 16$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} + 160 x - 16 \cdot 8}}{2 \cdot 4}$

Step 1.3.1.4.3

Multiply $- 16$ by $8$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} + 160 x - 128}}{2 \cdot 4}$

Step 1.3.1.5

Subtract $128$ from $576$ .

$y = \frac{- 24 \pm \sqrt{- 80 x^{2} + 160 x + 448}}{2 \cdot 4}$

Step 1.3.1.6

Factor $16$ out of $- 80 x^{2} + 160 x + 448$ .

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Step 1.3.1.6.1

Factor $16$ out of $- 80 x^{2}$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 160 x + 448}}{2 \cdot 4}$

Step 1.3.1.6.2

Factor $16$ out of $160 x$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 16 (10 x) + 448}}{2 \cdot 4}$

Step 1.3.1.6.3

Factor $16$ out of $448$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 16 (10 x) + 16 (28)}}{2 \cdot 4}$

Step 1.3.1.6.4

Factor $16$ out of $16 (- 5 x^{2}) + 16 (10 x)$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2} + 10 x) + 16 (28)}}{2 \cdot 4}$

Step 1.3.1.6.5

Factor $16$ out of $16 (- 5 x^{2} + 10 x) + 16 (28)$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.3.1.7

Rewrite $16 (- 5 x^{2} + 10 x + 28)$ as ${(2^{2})}^{2} (- 5 x^{2} + 10 x + 28)$ .

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Step 1.3.1.7.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 24 \pm \sqrt{4^{2} (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.3.1.7.2

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

$y = \frac{- 24 \pm \sqrt{{(2^{2})}^{2} (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.3.1.8

Pull terms out from under the radical.

$y = \frac{- 24 \pm 2^{2} \sqrt{- 5 x^{2} + 10 x + 28}}{2 \cdot 4}$

Step 1.3.1.9

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

$y = \frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{2 \cdot 4}$

Step 1.3.2

Multiply $2$ by $4$ .

$y = \frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{8}$

Step 1.3.3

Simplify $\frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{8}$ .

$y = \frac{- 6 \pm \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.4

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

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 4 \cdot (5 x^{2} - 10 x + 8)}}{2 \cdot 4}$

Step 1.4.1.2

Multiply $- 4$ by $4$ .

$y = \frac{- 24 \pm \sqrt{576 - 16 \cdot (5 x^{2} - 10 x + 8)}}{2 \cdot 4}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{576 - 16 (5 x^{2}) - 16 (- 10 x) - 16 \cdot 8}}{2 \cdot 4}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $5$ by $- 16$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} - 16 (- 10 x) - 16 \cdot 8}}{2 \cdot 4}$

Step 1.4.1.4.2

Multiply $- 10$ by $- 16$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} + 160 x - 16 \cdot 8}}{2 \cdot 4}$

Step 1.4.1.4.3

Multiply $- 16$ by $8$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} + 160 x - 128}}{2 \cdot 4}$

Step 1.4.1.5

Subtract $128$ from $576$ .

$y = \frac{- 24 \pm \sqrt{- 80 x^{2} + 160 x + 448}}{2 \cdot 4}$

Step 1.4.1.6

Factor $16$ out of $- 80 x^{2} + 160 x + 448$ .

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Step 1.4.1.6.1

Factor $16$ out of $- 80 x^{2}$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 160 x + 448}}{2 \cdot 4}$

Step 1.4.1.6.2

Factor $16$ out of $160 x$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 16 (10 x) + 448}}{2 \cdot 4}$

Step 1.4.1.6.3

Factor $16$ out of $448$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 16 (10 x) + 16 (28)}}{2 \cdot 4}$

Step 1.4.1.6.4

Factor $16$ out of $16 (- 5 x^{2}) + 16 (10 x)$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2} + 10 x) + 16 (28)}}{2 \cdot 4}$

Step 1.4.1.6.5

Factor $16$ out of $16 (- 5 x^{2} + 10 x) + 16 (28)$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.4.1.7

Rewrite $16 (- 5 x^{2} + 10 x + 28)$ as ${(2^{2})}^{2} (- 5 x^{2} + 10 x + 28)$ .

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Step 1.4.1.7.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 24 \pm \sqrt{4^{2} (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.4.1.7.2

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

$y = \frac{- 24 \pm \sqrt{{(2^{2})}^{2} (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.4.1.8

Pull terms out from under the radical.

$y = \frac{- 24 \pm 2^{2} \sqrt{- 5 x^{2} + 10 x + 28}}{2 \cdot 4}$

Step 1.4.1.9

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

$y = \frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{2 \cdot 4}$

Step 1.4.2

Multiply $2$ by $4$ .

$y = \frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{8}$

Step 1.4.3

Simplify $\frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{8}$ .

$y = \frac{- 6 \pm \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{- 6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.4.5

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.4.6

Factor $- 1$ out of $\sqrt{- 5 x^{2} + 10 x + 28}$ .

$y = \frac{- 1 \cdot 6 - 1 (- \sqrt{- 5 x^{2} + 10 x + 28})}{2}$

Step 1.4.7

Factor $- 1$ out of $- 1 (6) - 1 (- \sqrt{- 5 x^{2} + 10 x + 28})$ .

$y = \frac{- 1 (6 - \sqrt{- 5 x^{2} + 10 x + 28})}{2}$

Step 1.4.8

Move the negative in front of the fraction.

$y = - \frac{6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 4 \cdot (5 x^{2} - 10 x + 8)}}{2 \cdot 4}$

Step 1.5.1.2

Multiply $- 4$ by $4$ .

$y = \frac{- 24 \pm \sqrt{576 - 16 \cdot (5 x^{2} - 10 x + 8)}}{2 \cdot 4}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{576 - 16 (5 x^{2}) - 16 (- 10 x) - 16 \cdot 8}}{2 \cdot 4}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $5$ by $- 16$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} - 16 (- 10 x) - 16 \cdot 8}}{2 \cdot 4}$

Step 1.5.1.4.2

Multiply $- 10$ by $- 16$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} + 160 x - 16 \cdot 8}}{2 \cdot 4}$

Step 1.5.1.4.3

Multiply $- 16$ by $8$ .

$y = \frac{- 24 \pm \sqrt{576 - 80 x^{2} + 160 x - 128}}{2 \cdot 4}$

Step 1.5.1.5

Subtract $128$ from $576$ .

$y = \frac{- 24 \pm \sqrt{- 80 x^{2} + 160 x + 448}}{2 \cdot 4}$

Step 1.5.1.6

Factor $16$ out of $- 80 x^{2} + 160 x + 448$ .

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Step 1.5.1.6.1

Factor $16$ out of $- 80 x^{2}$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 160 x + 448}}{2 \cdot 4}$

Step 1.5.1.6.2

Factor $16$ out of $160 x$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 16 (10 x) + 448}}{2 \cdot 4}$

Step 1.5.1.6.3

Factor $16$ out of $448$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2}) + 16 (10 x) + 16 (28)}}{2 \cdot 4}$

Step 1.5.1.6.4

Factor $16$ out of $16 (- 5 x^{2}) + 16 (10 x)$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2} + 10 x) + 16 (28)}}{2 \cdot 4}$

Step 1.5.1.6.5

Factor $16$ out of $16 (- 5 x^{2} + 10 x) + 16 (28)$ .

$y = \frac{- 24 \pm \sqrt{16 (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.5.1.7

Rewrite $16 (- 5 x^{2} + 10 x + 28)$ as ${(2^{2})}^{2} (- 5 x^{2} + 10 x + 28)$ .

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Step 1.5.1.7.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 24 \pm \sqrt{4^{2} (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.5.1.7.2

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

$y = \frac{- 24 \pm \sqrt{{(2^{2})}^{2} (- 5 x^{2} + 10 x + 28)}}{2 \cdot 4}$

Step 1.5.1.8

Pull terms out from under the radical.

$y = \frac{- 24 \pm 2^{2} \sqrt{- 5 x^{2} + 10 x + 28}}{2 \cdot 4}$

Step 1.5.1.9

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

$y = \frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{2 \cdot 4}$

Step 1.5.2

Multiply $2$ by $4$ .

$y = \frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{8}$

Step 1.5.3

Simplify $\frac{- 24 \pm 4 \sqrt{- 5 x^{2} + 10 x + 28}}{8}$ .

$y = \frac{- 6 \pm \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{- 6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.5.5

Factor $- 1$ out of $- 6 - \sqrt{- 5 x^{2} + 10 x + 28}$ .

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Step 1.5.5.1

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.5.5.2

Factor $- 1$ out of $- \sqrt{- 5 x^{2} + 10 x + 28}$ .

$y = \frac{- 1 \cdot 6 - (\sqrt{- 5 x^{2} + 10 x + 28})}{2}$

Step 1.5.5.3

Factor $- 1$ out of $- 1 (6) - (\sqrt{- 5 x^{2} + 10 x + 28})$ .

$y = \frac{- 1 (6 + \sqrt{- 5 x^{2} + 10 x + 28})}{2}$

Step 1.5.5.4

Rewrite $- 1 (6 + \sqrt{- 5 x^{2} + 10 x + 28})$ as $- (6 + \sqrt{- 5 x^{2} + 10 x + 28})$ .

$y = \frac{- (6 + \sqrt{- 5 x^{2} + 10 x + 28})}{2}$

Step 1.5.6

Move the negative in front of the fraction.

$y = - \frac{6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 1.6

The final answer is the combination of both solutions.

$y = - \frac{6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

$y = - \frac{6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

$y = - \frac{6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

$y = - \frac{6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = - \frac{6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2} \to f (x) = - \frac{6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

$y = - \frac{6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2} \to f (x) = - \frac{6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$

Step 3

Because the $y$ variable in the equation $5 x^{2} + 4 y^{2} - 10 x + 24 y + 8 = 0$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 x^{2} + 4 y^{2} - 10 x + 24 y + 8) = \frac{d}{d x} (0)$

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

By the Sum Rule, the derivative of $5 x^{2} + 4 y^{2} - 10 x + 24 y + 8$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 3.2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$5 (2 x) + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.3

Evaluate $\frac{d}{d x} [4 y^{2}]$ .

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Step 3.2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 y^{2}$ with respect to $x$ is $4 \frac{d}{d x} [y^{2}]$ .

$10 x + 4 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.3.2.1

To apply the Chain Rule, set $u$ as $y$ .

$10 x + 4 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$10 x + 4 (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.3.2.3

Replace all occurrences of $u$ with $y$ .

$10 x + 4 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.3.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$10 x + 4 (2 y y') + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.3.4

Multiply $2$ by $4$ .

$10 x + 8 y y' + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.4

Evaluate $\frac{d}{d x} [- 10 x]$ .

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Step 3.2.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$10 x + 8 y y' - 10 \frac{d}{d x} [x] + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.4.2

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

$10 x + 8 y y' - 10 \cdot 1 + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.4.3

Multiply $- 10$ by $1$ .

$10 x + 8 y y' - 10 + \frac{d}{d x} [24 y] + \frac{d}{d x} [8]$

Step 3.2.5

Evaluate $\frac{d}{d x} [24 y]$ .

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Step 3.2.5.1

Since $24$ is constant with respect to $x$ , the derivative of $24 y$ with respect to $x$ is $24 \frac{d}{d x} [y]$ .

$10 x + 8 y y' - 10 + 24 \frac{d}{d x} [y] + \frac{d}{d x} [8]$

Step 3.2.5.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$10 x + 8 y y' - 10 + 24 y' + \frac{d}{d x} [8]$

Step 3.2.6

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$10 x + 8 y y' - 10 + 24 y' + 0$

Step 3.2.7

Simplify.

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Step 3.2.7.1

Add $10 x + 8 y y' - 10 + 24 y'$ and $0$ .

$10 x + 8 y y' - 10 + 24 y'$

Step 3.2.7.2

Reorder terms.

$8 y y' + 10 x + 24 y' - 10$

Step 3.3

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

$0$

Step 3.4

Reform the equation by setting the left side equal to the right side.

$8 y y' + 10 x + 24 y' - 10 = 0$

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 3.5.1.1

Subtract $10 x$ from both sides of the equation.

$8 y y' + 24 y' - 10 = - 10 x$

Step 3.5.1.2

Add $10$ to both sides of the equation.

$8 y y' + 24 y' = - 10 x + 10$

Step 3.5.2

Factor $8 y'$ out of $8 y y' + 24 y'$ .

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Step 3.5.2.1

Factor $8 y'$ out of $8 y y'$ .

$8 y' y + 24 y' = - 10 x + 10$

Step 3.5.2.2

Factor $8 y'$ out of $24 y'$ .

$8 y' y + 8 y' \cdot 3 = - 10 x + 10$

Step 3.5.2.3

Factor $8 y'$ out of $8 y' y + 8 y' \cdot 3$ .

$8 y' (y + 3) = - 10 x + 10$

Step 3.5.3

Divide each term in $8 y' (y + 3) = - 10 x + 10$ by $8 (y + 3)$ and simplify.

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Step 3.5.3.1

Divide each term in $8 y' (y + 3) = - 10 x + 10$ by $8 (y + 3)$ .

$\frac{8 y' (y + 3)}{8 (y + 3)} = \frac{- 10 x}{8 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $8$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{8 y' (y + 3)}{8 (y + 3)} = \frac{- 10 x}{8 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.2.1.2

Rewrite the expression.

$\frac{y' (y + 3)}{y + 3} = \frac{- 10 x}{8 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.2.2

Cancel the common factor of $y + 3$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

$\frac{y' (y + 3)}{y + 3} = \frac{- 10 x}{8 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 10 x}{8 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Simplify each term.

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Step 3.5.3.3.1.1

Cancel the common factor of $- 10$ and $8$ .

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Step 3.5.3.3.1.1.1

Factor $2$ out of $- 10 x$ .

$y' = \frac{2 (- 5 x)}{8 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.3.1.1.2

Cancel the common factors.

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Step 3.5.3.3.1.1.2.1

Factor $2$ out of $8 (y + 3)$ .

$y' = \frac{2 (- 5 x)}{2 (4 (y + 3))} + \frac{10}{8 (y + 3)}$

Step 3.5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- 5 x)}{2 (4 (y + 3))} + \frac{10}{8 (y + 3)}$

Step 3.5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 5 x}{4 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{5 x}{4 (y + 3)} + \frac{10}{8 (y + 3)}$

Step 3.5.3.3.1.3

Cancel the common factor of $10$ and $8$ .

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Step 3.5.3.3.1.3.1

Factor $2$ out of $10$ .

$y' = - \frac{5 x}{4 (y + 3)} + \frac{2 (5)}{8 (y + 3)}$

Step 3.5.3.3.1.3.2

Cancel the common factors.

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Step 3.5.3.3.1.3.2.1

Factor $2$ out of $8 (y + 3)$ .

$y' = - \frac{5 x}{4 (y + 3)} + \frac{2 (5)}{2 (4 (y + 3))}$

Step 3.5.3.3.1.3.2.2

Cancel the common factor.

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

Step 3.5.3.3.1.3.2.3

Rewrite the expression.

$y' = - \frac{5 x}{4 (y + 3)} + \frac{5}{4 (y + 3)}$

Step 3.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{5 x}{4 (y + 3)} + \frac{5}{4 (y + 3)}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{5 x}{4 (y + 3)} + \frac{5}{4 (y + 3)} = 0$ .

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Step 4.1

Subtract $\frac{5}{4 (y + 3)}$ from both sides of the equation.

$- \frac{5 x}{4 (y + 3)} = - \frac{5}{4 (y + 3)}$

Step 4.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 5 x = - 5$

Step 4.3

Divide each term in $- 5 x = - 5$ by $- 5$ and simplify.

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Step 4.3.1

Divide each term in $- 5 x = - 5$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 5}{- 5}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 5$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 5}{- 5}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 5}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 5$ by $- 5$ .

$x = 1$

Step 5

Solve the function $f (x) = - \frac{6 - \sqrt{- 5 x^{2} + 10 x + 28}}{2}$ at $x = 1$ .

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{6 - \sqrt{- 5 {(1)}^{2} + 10 (1) + 28}}{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

One to any power is one.

$f (1) = - \frac{6 - \sqrt{- 5 \cdot 1 + 10 (1) + 28}}{2}$

Step 5.2.1.2

Multiply $- 5$ by $1$ .

$f (1) = - \frac{6 - \sqrt{- 5 + 10 (1) + 28}}{2}$

Step 5.2.1.3

Multiply $10$ by $1$ .

$f (1) = - \frac{6 - \sqrt{- 5 + 10 + 28}}{2}$

Step 5.2.1.4

Add $- 5$ and $10$ .

$f (1) = - \frac{6 - \sqrt{5 + 28}}{2}$

Step 5.2.1.5

Add $5$ and $28$ .

$f (1) = - \frac{6 - \sqrt{33}}{2}$

Step 5.2.2

The final answer is $- \frac{6 - \sqrt{33}}{2}$ .

$- \frac{6 - \sqrt{33}}{2}$

Step 6

Solve the function $f (x) = - \frac{6 + \sqrt{- 5 x^{2} + 10 x + 28}}{2}$ at $x = 1$ .

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{6 + \sqrt{- 5 {(1)}^{2} + 10 (1) + 28}}{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

One to any power is one.

$f (1) = - \frac{6 + \sqrt{- 5 \cdot 1 + 10 (1) + 28}}{2}$

Step 6.2.1.2

Multiply $- 5$ by $1$ .

$f (1) = - \frac{6 + \sqrt{- 5 + 10 (1) + 28}}{2}$

Step 6.2.1.3

Multiply $10$ by $1$ .

$f (1) = - \frac{6 + \sqrt{- 5 + 10 + 28}}{2}$

Step 6.2.1.4

Add $- 5$ and $10$ .

$f (1) = - \frac{6 + \sqrt{5 + 28}}{2}$

Step 6.2.1.5

Add $5$ and $28$ .

$f (1) = - \frac{6 + \sqrt{33}}{2}$

Step 6.2.2

The final answer is $- \frac{6 + \sqrt{33}}{2}$ .

$- \frac{6 + \sqrt{33}}{2}$

Step 7

The horizontal tangent lines are $y = - \frac{6 - \sqrt{33}}{2}, y = - \frac{6 + \sqrt{33}}{2}$

$y = - \frac{6 - \sqrt{33}}{2}, y = - \frac{6 + \sqrt{33}}{2}$

Step 8