Finite Math Examples

$y = \sqrt{64 - x^{2}}$ domain

Step 1

Rewrite the equation as $\sqrt{64 - x^{2}} = y$ .

$\sqrt{64 - x^{2}} = y$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{64 - x^{2}}}^{2} = y^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 - x^{2}}$ as ${(64 - x^{2})}^{\frac{1}{2}}$ .

${({(64 - x^{2})}^{\frac{1}{2}})}^{2} = y^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(64 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(64 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(64 - x^{2})}^{\frac{1}{2} \cdot 2} = y^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(64 - x^{2})}^{\frac{1}{2} \cdot 2} = y^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(64 - x^{2})}^{1} = y^{2}$

Step 3.2.1.2

Simplify.

$64 - x^{2} = y^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $64$ from both sides of the equation.

$- x^{2} = y^{2} - 64$

Step 4.2

Divide each term in $- x^{2} = y^{2} - 64$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x^{2} = y^{2} - 64$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{y^{2}}{- 1} + \frac{- 64}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{y^{2}}{- 1} + \frac{- 64}{- 1}$

Step 4.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{y^{2}}{- 1} + \frac{- 64}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Move the negative one from the denominator of $\frac{y^{2}}{- 1}$ .

$x^{2} = - 1 \cdot y^{2} + \frac{- 64}{- 1}$

Step 4.2.3.1.2

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

$x^{2} = - y^{2} + \frac{- 64}{- 1}$

Step 4.2.3.1.3

Divide $- 64$ by $- 1$ .

$x^{2} = - y^{2} + 64$

Step 4.3

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

$x = \pm \sqrt{- y^{2} + 64}$

Step 4.4

Simplify $\pm \sqrt{- y^{2} + 64}$ .

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Step 4.4.1

Simplify the expression.

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Step 4.4.1.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{- y^{2} + 8^{2}}$

Step 4.4.1.2

Reorder $- y^{2}$ and $8^{2}$ .

$x = \pm \sqrt{8^{2} - y^{2}}$

Step 4.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = y$ .

$x = \pm \sqrt{(8 + y) (8 - y)}$

Step 4.5

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

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Step 4.5.1

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

$x = \sqrt{(8 + y) (8 - y)}$

Step 4.5.2

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

$x = - \sqrt{(8 + y) (8 - y)}$

Step 4.5.3

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

$x = \sqrt{(8 + y) (8 - y)}$

$x = - \sqrt{(8 + y) (8 - y)}$

$x = \sqrt{(8 + y) (8 - y)}$

$x = - \sqrt{(8 + y) (8 - y)}$

$x = \sqrt{(8 + y) (8 - y)}$

$x = - \sqrt{(8 + y) (8 - y)}$