Precalculus Examples

$y^{2} = 4 - x^{2}$

Step 1

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

$y = \pm \sqrt{4 - x^{2}}$

Step 2

Simplify $\pm \sqrt{4 - x^{2}}$ .

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Step 2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

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

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

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

Step 4

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.

$f (x) = \sqrt{(2 + x) (2 - x)}, - \sqrt{(2 + x) (2 - x)}$