Solve Using the Square Root Property (x-1)^2=-9

${(x - 1)}^{2} = - 9$

Step 1

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

$x - 1 = \pm \sqrt{- 9}$

Step 2

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$x - 1 = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x - 1 = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$x - 1 = \pm i \cdot \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$x - 1 = \pm i \cdot \sqrt{3^{2}}$

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

$x - 1 = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$x - 1 = \pm 3 i$

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.

$x - 1 = 3 i$

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

Add $1$ to both sides of the equation.

$x = 3 i + 1$

Reorder $3 i$ and $1$ .

$x = 1 + 3 i$

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

$x - 1 = - 3 i$

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

Add $1$ to both sides of the equation.

$x = - 3 i + 1$

Reorder $- 3 i$ and $1$ .

$x = 1 - 3 i$

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

$x = 1 + 3 i, 1 - 3 i$