Precalculus Examples

$x^{2} + y^{2} = 36$ , $y^{2} - 2 x = 36$

Step 1

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

$y^{2} = 36 - x^{2}$

$y^{2} - 2 x = 36$

Step 2

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

$y = \pm \sqrt{36 - x^{2}}$

$y^{2} - 2 x = 36$

Step 3

Simplify $\pm \sqrt{36 - x^{2}}$ .

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Rewrite $36$ as $6^{2}$ .

$y = \pm \sqrt{6^{2} - x^{2}}$

$y^{2} - 2 x = 36$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$y = \pm \sqrt{(6 + x) (6 - x)}$

$y^{2} - 2 x = 36$

$y = \pm \sqrt{(6 + x) (6 - x)}$

$y^{2} - 2 x = 36$

Step 4

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

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

$y = \sqrt{(6 + x) (6 - x)}$

$y^{2} - 2 x = 36$

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

$y = - \sqrt{(6 + x) (6 - x)}$

$y^{2} - 2 x = 36$

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

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

$y^{2} - 2 x = 36$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

$y^{2} - 2 x = 36$

Step 5

Add $2 x$ to both sides of the equation.

$y^{2} = 36 + 2 x$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

Step 6

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

$y = \pm \sqrt{36 + 2 x}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

Step 7

Factor $2$ out of $36 + 2 x$ .

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Factor $2$ out of $36$ .

$y = \pm \sqrt{2 \cdot 18 + 2 x}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

Factor $2$ out of $2 \cdot 18 + 2 x$ .

$y = \pm \sqrt{2 (18 + x)}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

$y = \pm \sqrt{2 (18 + x)}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

Step 8

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

Tap for more steps...

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

$y = \sqrt{2 (18 + x)}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

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

$y = - \sqrt{2 (18 + x)}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

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

$y = \sqrt{2 (18 + x)}$

$y = - \sqrt{2 (18 + x)}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

$y = \sqrt{2 (18 + x)}$

$y = - \sqrt{2 (18 + x)}$

$y = \sqrt{(6 + x) (6 - x)}$

$y = - \sqrt{(6 + x) (6 - x)}$

Step 9

Create a graph to locate the intersection of the equations. The intersection of the system of equations is the solution.

$(0, 6)$

$(0, - 6)$

$(- 2, 4 \sqrt{2})$

$(- 2, - 4 \sqrt{2})$

Step 10