Precalculus Examples

$x^{2} + y^{2} = 25$

Step 1

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

$x^{2} = 25 - y^{2}$

Step 2

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

$x = \pm \sqrt{25 - y^{2}}$

Step 3

Simplify $\pm \sqrt{25 - y^{2}}$ .

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} - y^{2}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = y$ .

$x = \pm \sqrt{(5 + y) (5 - y)}$

Step 4

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

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Step 4.1

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

$x = \sqrt{(5 + y) (5 - y)}$

Step 4.2

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

$x = - \sqrt{(5 + y) (5 - y)}$

Step 4.3

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

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 5

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

$f (y) = \sqrt{(5 + y) (5 - y)}, - \sqrt{(5 + y) (5 - y)}$