Calculus Examples

$x^{3} + 3 x^{2} y + y^{3} = 8$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} + 3 x^{2} y + y^{3}) = \frac{d}{d x} (8)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} + 3 x^{2} y + y^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2} y] + \frac{d}{d x} [y^{3}]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2} y] + \frac{d}{d x} [y^{3}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [3 x^{2} y] + \frac{d}{d x} [y^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{2} y]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} y$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} y]$ .

$3 x^{2} + 3 \frac{d}{d x} [x^{2} y] + \frac{d}{d x} [y^{3}]$

Step 2.2.2

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 $f (x) = x^{2}$ and $g (x) = y$ .

$3 x^{2} + 3 (x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [y^{3}]$

Step 2.2.3

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

$3 x^{2} + 3 (x^{2} y' + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [y^{3}]$

Step 2.2.4

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

$3 x^{2} + 3 (x^{2} y' + y (2 x)) + \frac{d}{d x} [y^{3}]$

Step 2.2.5

Move $2$ to the left of $y$ .

$3 x^{2} + 3 (x^{2} y' + 2 y x) + \frac{d}{d x} [y^{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [y^{3}]$ .

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.1.1

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

$3 x^{2} + 3 (x^{2} y' + 2 y x) + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

Step 2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 x^{2} + 3 (x^{2} y' + 2 y x) + 3 u^{2} \frac{d}{d x} [y]$

Step 2.3.1.3

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

$3 x^{2} + 3 (x^{2} y' + 2 y x) + 3 y^{2} \frac{d}{d x} [y]$

Step 2.3.2

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

$3 x^{2} + 3 (x^{2} y' + 2 y x) + 3 y^{2} y'$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$3 x^{2} + 3 (x^{2} y') + 3 (2 y x) + 3 y^{2} y'$

Step 2.4.2

Multiply $2$ by $3$ .

$3 x^{2} + 3 x^{2} y' + 6 y x + 3 y^{2} y'$

Step 2.4.3

Reorder terms.

$3 x^{2} + 3 x^{2} y' + 6 x y + 3 y^{2} y'$

Step 3

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

$0$

Step 4

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

$3 x^{2} + 3 x^{2} y' + 6 x y + 3 y^{2} y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $3 x^{2}$ from both sides of the equation.

$3 x^{2} y' + 6 x y + 3 y^{2} y' = - 3 x^{2}$

Step 5.1.2

Subtract $6 x y$ from both sides of the equation.

$3 x^{2} y' + 3 y^{2} y' = - 3 x^{2} - 6 x y$

Step 5.2

Factor $3 y'$ out of $3 x^{2} y' + 3 y^{2} y'$ .

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Step 5.2.1

Factor $3 y'$ out of $3 x^{2} y'$ .

$3 y' x^{2} + 3 y^{2} y' = - 3 x^{2} - 6 x y$

Step 5.2.2

Factor $3 y'$ out of $3 y^{2} y'$ .

$3 y' x^{2} + 3 y' y^{2} = - 3 x^{2} - 6 x y$

Step 5.2.3

Factor $3 y'$ out of $3 y' x^{2} + 3 y' y^{2}$ .

$3 y' (x^{2} + y^{2}) = - 3 x^{2} - 6 x y$

Step 5.3

Divide each term in $3 y' (x^{2} + y^{2}) = - 3 x^{2} - 6 x y$ by $3 (x^{2} + y^{2})$ and simplify.

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Step 5.3.1

Divide each term in $3 y' (x^{2} + y^{2}) = - 3 x^{2} - 6 x y$ by $3 (x^{2} + y^{2})$ .

$\frac{3 y' (x^{2} + y^{2})}{3 (x^{2} + y^{2})} = \frac{- 3 x^{2}}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 y' (x^{2} + y^{2})}{3 (x^{2} + y^{2})} = \frac{- 3 x^{2}}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (x^{2} + y^{2})}{x^{2} + y^{2}} = \frac{- 3 x^{2}}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.2.2

Cancel the common factor of $x^{2} + y^{2}$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (x^{2} + y^{2})}{x^{2} + y^{2}} = \frac{- 3 x^{2}}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 x^{2}}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 5.3.3.1.1.1

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

$y' = \frac{3 (- x^{2})}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

$y' = \frac{3 (- x^{2})}{3 (x^{2} + y^{2})} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- x^{2}}{x^{2} + y^{2}} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{x^{2}}{x^{2} + y^{2}} + \frac{- 6 x y}{3 (x^{2} + y^{2})}$

Step 5.3.3.1.3

Cancel the common factor of $- 6$ and $3$ .

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Step 5.3.3.1.3.1

Factor $3$ out of $- 6 x y$ .

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

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

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

Step 5.3.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{x^{2}}{x^{2} + y^{2}} + \frac{- 2 x y}{x^{2} + y^{2}}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{x^{2}}{x^{2} + y^{2}} - \frac{2 x y}{x^{2} + y^{2}}$

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- x^{2} - 2 x y}{x^{2} + y^{2}}$

Step 5.3.3.2.2

Factor $x$ out of $- x^{2} - 2 x y$ .

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Step 5.3.3.2.2.1

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

$y' = \frac{x (- x) - 2 x y}{x^{2} + y^{2}}$

Step 5.3.3.2.2.2

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

$y' = \frac{x (- x) + x (- 2 y)}{x^{2} + y^{2}}$

Step 5.3.3.2.2.3

Factor $x$ out of $x (- x) + x (- 2 y)$ .

$y' = \frac{x (- x - 2 y)}{x^{2} + y^{2}}$

Step 5.3.3.2.3

Factor $- 1$ out of $- x$ .

$y' = \frac{x (- (x) - 2 y)}{x^{2} + y^{2}}$

Step 5.3.3.2.4

Factor $- 1$ out of $- 2 y$ .

$y' = \frac{x (- (x) - (2 y))}{x^{2} + y^{2}}$

Step 5.3.3.2.5

Factor $- 1$ out of $- (x) - (2 y)$ .

$y' = \frac{x (- (x + 2 y))}{x^{2} + y^{2}}$

Step 5.3.3.2.6

Simplify the expression.

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Step 5.3.3.2.6.1

Rewrite $- (x + 2 y)$ as $- 1 (x + 2 y)$ .

$y' = \frac{x (- 1 (x + 2 y))}{x^{2} + y^{2}}$

Step 5.3.3.2.6.2

Move the negative in front of the fraction.

$y' = - \frac{x (x + 2 y)}{x^{2} + y^{2}}$

Step 6

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

$\frac{d y}{d x} = - \frac{x (x + 2 y)}{x^{2} + y^{2}}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$x (x + 2 y) = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 2 y = 0$

Step 7.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 7.2.3

Set $x + 2 y$ equal to $0$ and solve for $x$ .

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Step 7.2.3.1

Set $x + 2 y$ equal to $0$ .

$x + 2 y = 0$

Step 7.2.3.2

Subtract $2 y$ from both sides of the equation.

$x = - 2 y$

Step 7.2.4

The final solution is all the values that make $x (x + 2 y) = 0$ true.

$x = 0$

$x = - 2 y$

$x = 0$

$x = - 2 y$

$x = 0$

$x = - 2 y$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $0^{3} + 3 \cdot (0^{2} y) + y^{3}$ .

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Raising $0$ to any positive power yields $0$ .

$0 + 3 \cdot (0^{2} y) + y^{3} = 8$

Step 8.1.1.2

Raising $0$ to any positive power yields $0$ .

$0 + 3 \cdot (0 y) + y^{3} = 8$

Step 8.1.1.3

Multiply $0$ by $y$ .

$0 + 3 \cdot 0 + y^{3} = 8$

Step 8.1.1.4

Multiply $3$ by $0$ .

$0 + 0 + y^{3} = 8$

Step 8.1.2

Combine the opposite terms in $0 + 0 + y^{3}$ .

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Step 8.1.2.1

Add $0$ and $0$ .

$0 + y^{3} = 8$

Step 8.1.2.2

Add $0$ and $y^{3}$ .

$y^{3} = 8$

Step 8.2

Subtract $8$ from both sides of the equation.

$y^{3} - 8 = 0$

Step 8.3

Factor the left side of the equation.

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Step 8.3.1

Rewrite $8$ as $2^{3}$ .

$y^{3} - 2^{3} = 0$

Step 8.3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 2$ .

$(y - 2) (y^{2} + y \cdot 2 + 2^{2}) = 0$

Step 8.3.3

Simplify.

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Step 8.3.3.1

Move $2$ to the left of $y$ .

$(y - 2) (y^{2} + 2 y + 2^{2}) = 0$

Step 8.3.3.2

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

$(y - 2) (y^{2} + 2 y + 4) = 0$

Step 8.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 2 = 0$

$y^{2} + 2 y + 4 = 0$

Step 8.5

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 8.5.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 8.5.2

Add $2$ to both sides of the equation.

$y = 2$

Step 8.6

Set $y^{2} + 2 y + 4$ equal to $0$ and solve for $y$ .

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Step 8.6.1

Set $y^{2} + 2 y + 4$ equal to $0$ .

$y^{2} + 2 y + 4 = 0$

Step 8.6.2

Solve $y^{2} + 2 y + 4 = 0$ for $y$ .

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Step 8.6.2.1

Use the quadratic formula to find the solutions.

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

Step 8.6.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 8.6.2.3

Simplify.

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Step 8.6.2.3.1

Simplify the numerator.

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Step 8.6.2.3.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.6.2.3.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 8.6.2.3.1.3

Subtract $16$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 8.6.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 8.6.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.6.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.6.2.3.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 8.6.2.3.1.7.2

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

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 8.6.2.3.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 8.6.2.3.1.9

Move $2$ to the left of $i$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 8.6.2.3.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 8.6.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$y = - 1 \pm i \sqrt{3}$

Step 8.6.2.4

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

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Step 8.6.2.4.1

Simplify the numerator.

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Step 8.6.2.4.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.6.2.4.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 8.6.2.4.1.3

Subtract $16$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 8.6.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 8.6.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.6.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.6.2.4.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 8.6.2.4.1.7.2

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

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 8.6.2.4.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 8.6.2.4.1.9

Move $2$ to the left of $i$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 8.6.2.4.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 8.6.2.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$y = - 1 \pm i \sqrt{3}$

Step 8.6.2.4.4

Change the $\pm$ to $+$ .

$y = - 1 + i \sqrt{3}$

Step 8.6.2.5

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

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Step 8.6.2.5.1

Simplify the numerator.

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Step 8.6.2.5.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.6.2.5.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 8.6.2.5.1.3

Subtract $16$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 8.6.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 8.6.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.6.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.6.2.5.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 8.6.2.5.1.7.2

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

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 8.6.2.5.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 8.6.2.5.1.9

Move $2$ to the left of $i$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 8.6.2.5.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 8.6.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$y = - 1 \pm i \sqrt{3}$

Step 8.6.2.5.4

Change the $\pm$ to $-$ .

$y = - 1 - i \sqrt{3}$

Step 8.6.2.6

The final answer is the combination of both solutions.

$y = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 8.7

The final solution is all the values that make $(y - 2) (y^{2} + 2 y + 4) = 0$ true.

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 9

Solve for $y$ .

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Step 9.1

Simplify ${(- 2 y)}^{3} + 3 {(- 2 y)}^{2} y + y^{3}$ .

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Step 9.1.1

Simplify each term.

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Step 9.1.1.1

Apply the product rule to $- 2 y$ .

${(- 2)}^{3} y^{3} + 3 {(- 2 y)}^{2} y + y^{3} = 8$

Step 9.1.1.2

Raise $- 2$ to the power of $3$ .

$- 8 y^{3} + 3 {(- 2 y)}^{2} y + y^{3} = 8$

Step 9.1.1.3

Apply the product rule to $- 2 y$ .

$- 8 y^{3} + 3 ({(- 2)}^{2} y^{2}) y + y^{3} = 8$

Step 9.1.1.4

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 9.1.1.4.1

Move $y$ .

$- 8 y^{3} + 3 ({(- 2)}^{2} (y \cdot y^{2})) + y^{3} = 8$

Step 9.1.1.4.2

Multiply $y$ by $y^{2}$ .

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Step 9.1.1.4.2.1

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

$- 8 y^{3} + 3 ({(- 2)}^{2} (y^{1} y^{2})) + y^{3} = 8$

Step 9.1.1.4.2.2

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

$- 8 y^{3} + 3 ({(- 2)}^{2} y^{1 + 2}) + y^{3} = 8$

Step 9.1.1.4.3

Add $1$ and $2$ .

$- 8 y^{3} + 3 ({(- 2)}^{2} y^{3}) + y^{3} = 8$

Step 9.1.1.5

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

$- 8 y^{3} + 3 (4 y^{3}) + y^{3} = 8$

Step 9.1.1.6

Multiply $4$ by $3$ .

$- 8 y^{3} + 12 y^{3} + y^{3} = 8$

Step 9.1.2

Simplify by adding terms.

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Step 9.1.2.1

Add $- 8 y^{3}$ and $12 y^{3}$ .

$4 y^{3} + y^{3} = 8$

Step 9.1.2.2

Add $4 y^{3}$ and $y^{3}$ .

$5 y^{3} = 8$

Step 9.2

Divide each term in $5 y^{3} = 8$ by $5$ and simplify.

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Step 9.2.1

Divide each term in $5 y^{3} = 8$ by $5$ .

$\frac{5 y^{3}}{5} = \frac{8}{5}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $5$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{5 y^{3}}{5} = \frac{8}{5}$

Step 9.2.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{8}{5}$

Step 9.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{\frac{8}{5}}$

Step 9.4

Simplify $\sqrt[3]{\frac{8}{5}}$ .

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Step 9.4.1

Rewrite $\sqrt[3]{\frac{8}{5}}$ as $\frac{\sqrt[3]{8}}{\sqrt[3]{5}}$ .

$y = \frac{\sqrt[3]{8}}{\sqrt[3]{5}}$

Step 9.4.2

Simplify the numerator.

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Step 9.4.2.1

Rewrite $8$ as $2^{3}$ .

$y = \frac{\sqrt[3]{2^{3}}}{\sqrt[3]{5}}$

Step 9.4.2.2

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

$y = \frac{2}{\sqrt[3]{5}}$

Step 9.4.3

Multiply $\frac{2}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = \frac{2}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 9.4.4

Combine and simplify the denominator.

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Step 9.4.4.1

Multiply $\frac{2}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = \frac{2 {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 9.4.4.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$y = \frac{2 {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 9.4.4.3

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

$y = \frac{2 {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 9.4.4.4

Add $1$ and $2$ .

$y = \frac{2 {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 9.4.4.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 9.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$y = \frac{2 {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 9.4.4.5.2

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

$y = \frac{2 {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 9.4.4.5.3

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

$y = \frac{2 {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 9.4.4.5.4

Cancel the common factor of $3$ .

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Step 9.4.4.5.4.1

Cancel the common factor.

$y = \frac{2 {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 9.4.4.5.4.2

Rewrite the expression.

$y = \frac{2 {\sqrt[3]{5}}^{2}}{5^{1}}$

Step 9.4.4.5.5

Evaluate the exponent.

$y = \frac{2 {\sqrt[3]{5}}^{2}}{5}$

Step 9.4.5

Simplify the numerator.

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Step 9.4.5.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$y = \frac{2 \sqrt[3]{5^{2}}}{5}$

Step 9.4.5.2

Raise $5$ to the power of $2$ .

$y = \frac{2 \sqrt[3]{25}}{5}$

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 2)$

$(0, - 1 + i \sqrt{3})$

$(0, - 1 - i \sqrt{3})$

$(- 2 y, \frac{2 \sqrt[3]{25}}{5})$

Step 11