Pre-Algebra Examples

$x^{2} = - 32$

Step 1

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

$x = \pm \sqrt{- 32}$

Step 2

Simplify $\pm \sqrt{- 32}$ .

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Step 2.1

Rewrite $- 32$ as $- 1 (32)$ .

$x = \pm \sqrt{- 1 (32)}$

Step 2.2

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{32}$

Step 2.3

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

$x = \pm i \cdot \sqrt{32}$

Step 2.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.4.1

Factor $16$ out of $32$ .

$x = \pm i \cdot \sqrt{16 (2)}$

Step 2.4.2

Rewrite $16$ as $4^{2}$ .

$x = \pm i \cdot \sqrt{4^{2} \cdot 2}$

Step 2.5

Pull terms out from under the radical.

$x = \pm i \cdot (4 \sqrt{2})$

Step 2.6

Move $4$ to the left of $i$ .

$x = \pm 4 i \sqrt{2}$

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.

$x = 4 i \sqrt{2}$

Step 3.2

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

$x = - 4 i \sqrt{2}$

Step 3.3

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

$x = 4 i \sqrt{2}, - 4 i \sqrt{2}$