Pre-Algebra Examples

$12^{2} - 13^{2} = x^{2}$

Step 1

Rewrite the equation as $x^{2} = 12^{2} - 13^{2}$ .

$x^{2} = 12^{2} - 13^{2}$

Step 2

Raise $13$ to the power of $2$ .

$x^{2} = 12^{2} - 1 \cdot 169$

Step 3

Multiply $- 1$ by $169$ .

$x^{2} = 12^{2} - 169$

Step 4

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

$x = \pm \sqrt{12^{2} - 169}$

Step 5

Simplify $\pm \sqrt{12^{2} - 169}$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Raise $12$ to the power of $2$ .

$x = \pm \sqrt{144 - 169}$

Step 5.1.2

Subtract $169$ from $144$ .

$x = \pm \sqrt{- 25}$

Step 5.2

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Step 5.3

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

Step 5.4

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{25}$

Step 5.5

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

$x = \pm i \cdot \sqrt{5^{2}}$

Step 5.6

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 5$

Step 5.7

Move $5$ to the left of $i$ .

$x = \pm 5 i$

Step 6

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

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Step 6.1

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

$x = 5 i$

Step 6.2

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

$x = - 5 i$

Step 6.3

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

$x = 5 i, - 5 i$