Precalculus Examples

$f (x) = {(x + 5)}^{2} + 8$

Step 1

Set ${(x + 5)}^{2} + 8$ equal to $0$ .

${(x + 5)}^{2} + 8 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $8$ from both sides of the equation.

${(x + 5)}^{2} = - 8$

Step 2.2

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

$x + 5 = \pm \sqrt{- 8}$

Step 2.3

Simplify $\pm \sqrt{- 8}$ .

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Step 2.3.1

Rewrite $- 8$ as $- 1 (8)$ .

$x + 5 = \pm \sqrt{- 1 (8)}$

Step 2.3.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x + 5 = \pm \sqrt{- 1} \cdot \sqrt{8}$

Step 2.3.3

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

$x + 5 = \pm i \cdot \sqrt{8}$

Step 2.3.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.4.1

Factor $4$ out of $8$ .

$x + 5 = \pm i \cdot \sqrt{4 (2)}$

Step 2.3.4.2

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

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

Step 2.3.5

Pull terms out from under the radical.

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

Step 2.3.6

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

$x + 5 = \pm 2 i \sqrt{2}$

Step 2.4

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

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Step 2.4.1

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

$x + 5 = 2 i \sqrt{2}$

Step 2.4.2

Subtract $5$ from both sides of the equation.

$x = 2 i \sqrt{2} - 5$

Step 2.4.3

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

$x + 5 = - 2 i \sqrt{2}$

Step 2.4.4

Subtract $5$ from both sides of the equation.

$x = - 2 i \sqrt{2} - 5$

Step 2.4.5

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

$x = 2 i \sqrt{2} - 5, - 2 i \sqrt{2} - 5$

Step 3