Solve Using the Square Root Property (x-10)^2=-121

${(x - 10)}^{2} = - 121$

Step 1

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

$x - 10 = \pm \sqrt{- 121}$

Step 2

Simplify $\pm \sqrt{- 121}$ .

Rewrite $- 121$ as $- 1 (121)$ .

$x - 10 = \pm \sqrt{- 1 (121)}$

Rewrite $\sqrt{- 1 (121)}$ as $\sqrt{- 1} \cdot \sqrt{121}$ .

$x - 10 = \pm \sqrt{- 1} \cdot \sqrt{121}$

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

$x - 10 = \pm i \cdot \sqrt{121}$

Rewrite $121$ as $11^{2}$ .

$x - 10 = \pm i \cdot \sqrt{11^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - 10 = \pm i \cdot 11$

Move $11$ to the left of $i$ .

$x - 10 = \pm 11 i$

Step 3

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

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

$x - 10 = 11 i$

Move all terms not containing $x$ to the right side of the equation.

Add $10$ to both sides of the equation.

$x = 11 i + 10$

Reorder $11 i$ and $10$ .

$x = 10 + 11 i$

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

$x - 10 = - 11 i$

Move all terms not containing $x$ to the right side of the equation.

Add $10$ to both sides of the equation.

$x = - 11 i + 10$

Reorder $- 11 i$ and $10$ .

$x = 10 - 11 i$

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

$x = 10 + 11 i, 10 - 11 i$