Calculus Examples

$\frac{4 (- 32 y^{4} x + 12 y^{3})}{{(- 4 x y + 1)}^{3}} = 0$

Step 1

Set the numerator equal to zero.

$4 (- 32 y^{4} x + 12 y^{3}) = 0$

Step 2

Solve the equation for $y$ .

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Step 2.1

Divide each term in $4 (- 32 y^{4} x + 12 y^{3}) = 0$ by $4$ and simplify.

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Step 2.1.1

Divide each term in $4 (- 32 y^{4} x + 12 y^{3}) = 0$ by $4$ .

$\frac{4 (- 32 y^{4} x + 12 y^{3})}{4} = \frac{0}{4}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{4 (- 32 y^{4} x + 12 y^{3})}{4} = \frac{0}{4}$

Step 2.1.2.1.2

Divide $- 32 y^{4} x + 12 y^{3}$ by $1$ .

$- 32 y^{4} x + 12 y^{3} = \frac{0}{4}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $4$ .

$- 32 y^{4} x + 12 y^{3} = 0$

Step 2.2

Factor $4 y^{3}$ out of $- 32 y^{4} x + 12 y^{3}$ .

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Step 2.2.1

Factor $4 y^{3}$ out of $- 32 y^{4} x$ .

$4 y^{3} (- 8 y x) + 12 y^{3} = 0$

Step 2.2.2

Factor $4 y^{3}$ out of $12 y^{3}$ .

$4 y^{3} (- 8 y x) + 4 y^{3} (3) = 0$

Step 2.2.3

Factor $4 y^{3}$ out of $4 y^{3} (- 8 y x) + 4 y^{3} (3)$ .

$4 y^{3} (- 8 y x + 3) = 0$

Step 2.3

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

$y^{3} = 0$

$- 8 y x + 3 = 0$

Step 2.4

Set $y^{3}$ equal to $0$ and solve for $y$ .

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Step 2.4.1

Set $y^{3}$ equal to $0$ .

$y^{3} = 0$

Step 2.4.2

Solve $y^{3} = 0$ for $y$ .

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Step 2.4.2.1

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

$y = \sqrt[3]{0}$

Step 2.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{3}$ .

$y = \sqrt[3]{0^{3}}$

Step 2.4.2.2.2

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

$y = 0$

Step 2.5

Set $- 8 y x + 3$ equal to $0$ and solve for $y$ .

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Step 2.5.1

Set $- 8 y x + 3$ equal to $0$ .

$- 8 y x + 3 = 0$

Step 2.5.2

Solve $- 8 y x + 3 = 0$ for $y$ .

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Step 2.5.2.1

Subtract $3$ from both sides of the equation.

$- 8 y x = - 3$

Step 2.5.2.2

Divide each term in $- 8 y x = - 3$ by $- 8 x$ and simplify.

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Step 2.5.2.2.1

Divide each term in $- 8 y x = - 3$ by $- 8 x$ .

$\frac{- 8 y x}{- 8 x} = \frac{- 3}{- 8 x}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $- 8$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 8 y x}{- 8 x} = \frac{- 3}{- 8 x}$

Step 2.5.2.2.2.1.2

Rewrite the expression.

$\frac{y x}{x} = \frac{- 3}{- 8 x}$

Step 2.5.2.2.2.2

Cancel the common factor of $x$ .

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Step 2.5.2.2.2.2.1

Cancel the common factor.

$\frac{y x}{x} = \frac{- 3}{- 8 x}$

Step 2.5.2.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 3}{- 8 x}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{3}{8 x}$

Step 2.6

The final solution is all the values that make $4 y^{3} (- 8 y x + 3) = 0$ true.

$y = 0$

$y = \frac{3}{8 x}$

$y = 0$

$y = \frac{3}{8 x}$

Step 3

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = 0, \frac{3}{8 x}$