Precalculus Examples

$f (x) = {(x - 2)}^{2} + 2$

Step 1

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

${(x - 2)}^{2} + 2 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the equation.

${(x - 2)}^{2} = - 2$

Step 2.2

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

$x - 2 = \pm \sqrt{- 2}$

Step 2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x - 2 = \pm \sqrt{- 1 (2)}$

Step 2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x - 2 = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 2.3.3

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

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

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.

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

Step 2.4.2

Add $2$ to both sides of the equation.

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

Step 2.4.3

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

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

Step 2.4.4

Add $2$ to both sides of the equation.

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

Step 2.4.5

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

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

Step 3