Precalculus Examples

$y = \frac{| x |}{\sqrt{c - x^{2}}}$

Step 1

Rewrite the equation as $\frac{| x |}{\sqrt{c - x^{2}}} = y$ .

$\frac{| x |}{\sqrt{c - x^{2}}} = y$

Step 2

Simplify both sides of the equation.

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Step 2.1

Multiply $\frac{| x |}{\sqrt{c - x^{2}}}$ by $\frac{\sqrt{c - x^{2}}}{\sqrt{c - x^{2}}}$ .

$\frac{| x |}{\sqrt{c - x^{2}}} \cdot \frac{\sqrt{c - x^{2}}}{\sqrt{c - x^{2}}} = y$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

Multiply $\frac{| x |}{\sqrt{c - x^{2}}}$ by $\frac{\sqrt{c - x^{2}}}{\sqrt{c - x^{2}}}$ .

$\frac{| x | \sqrt{c - x^{2}}}{\sqrt{c - x^{2}} \sqrt{c - x^{2}}} = y$

Step 2.2.2

Raise $\sqrt{c - x^{2}}$ to the power of $1$ .

$\frac{| x | \sqrt{c - x^{2}}}{{\sqrt{c - x^{2}}}^{1} \sqrt{c - x^{2}}} = y$

Step 2.2.3

Raise $\sqrt{c - x^{2}}$ to the power of $1$ .

$\frac{| x | \sqrt{c - x^{2}}}{{\sqrt{c - x^{2}}}^{1} {\sqrt{c - x^{2}}}^{1}} = y$

Step 2.2.4

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

$\frac{| x | \sqrt{c - x^{2}}}{{\sqrt{c - x^{2}}}^{1 + 1}} = y$

Step 2.2.5

Add $1$ and $1$ .

$\frac{| x | \sqrt{c - x^{2}}}{{\sqrt{c - x^{2}}}^{2}} = y$

Step 2.2.6

Rewrite ${\sqrt{c - x^{2}}}^{2}$ as $c - x^{2}$ .

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Step 2.2.6.1

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

$\frac{| x | \sqrt{c - x^{2}}}{{({(c - x^{2})}^{\frac{1}{2}})}^{2}} = y$

Step 2.2.6.2

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

$\frac{| x | \sqrt{c - x^{2}}}{{(c - x^{2})}^{\frac{1}{2} \cdot 2}} = y$

Step 2.2.6.3

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

$\frac{| x | \sqrt{c - x^{2}}}{{(c - x^{2})}^{\frac{2}{2}}} = y$

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

$\frac{| x | \sqrt{c - x^{2}}}{{(c - x^{2})}^{\frac{2}{2}}} = y$

Step 2.2.6.4.2

Rewrite the expression.

$\frac{| x | \sqrt{c - x^{2}}}{{(c - x^{2})}^{1}} = y$

Step 2.2.6.5

Simplify.

$\frac{| x | \sqrt{c - x^{2}}}{c - x^{2}} = y$

Step 3

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

$\frac{| x | {(c - x^{2})}^{\frac{1}{2}}}{c - x^{2}} = y$

Step 4

Reduce $\frac{| x | {(c - x^{2})}^{\frac{1}{2}}}{c - x^{2}}$ .

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Step 4.1

Factor $c - x^{2}$ out of $| x | {(c - x^{2})}^{\frac{1}{2}}$ .

$\frac{(c - x^{2}) (| x | {(c - x^{2})}^{- \frac{1}{2}})}{c - x^{2}} = y$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\frac{(c - x^{2}) (| x | {(c - x^{2})}^{- \frac{1}{2}})}{(c - x^{2}) \cdot 1} = y$

Step 4.2.2

Cancel the common factor.

$\frac{(c - x^{2}) (| x | {(c - x^{2})}^{- \frac{1}{2}})}{(c - x^{2}) \cdot 1} = y$

Step 4.2.3

Rewrite the expression.

$\frac{| x | {(c - x^{2})}^{- \frac{1}{2}}}{1} = y$

Step 4.2.4

Divide $| x | {(c - x^{2})}^{- \frac{1}{2}}$ by $1$ .

$| x | {(c - x^{2})}^{- \frac{1}{2}} = y$

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $| x | {(c - x^{2})}^{- \frac{1}{2}} = y$ by ${(c - x^{2})}^{- \frac{1}{2}}$ and simplify.

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Step 5.1.1

Divide each term in $| x | {(c - x^{2})}^{- \frac{1}{2}} = y$ by ${(c - x^{2})}^{- \frac{1}{2}}$ .

$\frac{| x | {(c - x^{2})}^{- \frac{1}{2}}}{{(c - x^{2})}^{- \frac{1}{2}}} = \frac{y}{{(c - x^{2})}^{- \frac{1}{2}}}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of ${(c - x^{2})}^{- \frac{1}{2}}$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{| x | {(c - x^{2})}^{- \frac{1}{2}}}{{(c - x^{2})}^{- \frac{1}{2}}} = \frac{y}{{(c - x^{2})}^{- \frac{1}{2}}}$

Step 5.1.2.1.2

Divide $| x |$ by $1$ .

$| x | = \frac{y}{{(c - x^{2})}^{- \frac{1}{2}}}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Move ${(c - x^{2})}^{- \frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$| x | = y {(c - x^{2})}^{\frac{1}{2}}$

Step 5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm y {(c - x^{2})}^{\frac{1}{2}}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 5.3.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 5.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x =$