Algebra Examples

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

Step 1

Subtract $5$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify $\pm \sqrt{- 5}$ .

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Step 3.1

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

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

Step 3.3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Add $5$ to both sides of the equation.

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

Step 4.3

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

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

Step 4.4

Add $5$ to both sides of the equation.

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

Step 4.5

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

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