Finite Math Examples

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

Step 1

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

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

Step 2

Divide each term in $- \sqrt{x^{2} + y^{2}} = r$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \sqrt{x^{2} + y^{2}} = r$ by $- 1$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

Divide $\sqrt{x^{2} + y^{2}}$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative one from the denominator of $\frac{r}{- 1}$ .

$\sqrt{x^{2} + y^{2}} = - 1 \cdot r$

Step 2.3.2

Rewrite $- 1 \cdot r$ as $- r$ .

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

Step 3

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

${\sqrt{x^{2} + y^{2}}}^{2} = {(- r)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

$x^{2} + y^{2} = {(- r)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(- r)}^{2}$ .

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Step 4.3.1.1

Apply the product rule to $- r$ .

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

Step 4.3.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} + y^{2} = 1 r^{2}$

Step 4.3.1.3

Multiply $r^{2}$ by $1$ .

$x^{2} + y^{2} = r^{2}$

Step 5

Solve for $y$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

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

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

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

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

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