Pre-Algebra Examples

${(z + 7)}^{2} = - 20$

Step 1

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

$z + 7 = \pm \sqrt{- 20}$

Step 2

Simplify $\pm \sqrt{- 20}$ .

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Step 2.1

Rewrite $- 20$ as $- 1 (20)$ .

$z + 7 = \pm \sqrt{- 1 (20)}$

Step 2.2

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$z + 7 = \pm \sqrt{- 1} \cdot \sqrt{20}$

Step 2.3

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

$z + 7 = \pm i \cdot \sqrt{20}$

Step 2.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.4.1

Factor $4$ out of $20$ .

$z + 7 = \pm i \cdot \sqrt{4 (5)}$

Step 2.4.2

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

$z + 7 = \pm i \cdot \sqrt{2^{2} \cdot 5}$

Step 2.5

Pull terms out from under the radical.

$z + 7 = \pm i \cdot (2 \sqrt{5})$

Step 2.6

Move $2$ to the left of $i$ .

$z + 7 = \pm 2 i \sqrt{5}$

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.

$z + 7 = 2 i \sqrt{5}$

Step 3.2

Subtract $7$ from both sides of the equation.

$z = 2 i \sqrt{5} - 7$

Step 3.3

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

$z + 7 = - 2 i \sqrt{5}$

Step 3.4

Subtract $7$ from both sides of the equation.

$z = - 2 i \sqrt{5} - 7$

Step 3.5

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

$z = 2 i \sqrt{5} - 7, - 2 i \sqrt{5} - 7$