Solve Using the Quadratic Formula (x-2)^2-4(x-2)-21=0

${(x - 2)}^{2} - 4 (x - 2) - 21 = 0$

Step 1

Simplify the left side.

Simplify ${(x - 2)}^{2} - 4 (x - 2) - 21$ .

Simplify each term.

Apply the distributive property.

${(x - 2)}^{2} - 4 x - 4 \cdot - 2 - 21 = 0$

Multiply $- 4$ by $- 2$ .

${(x - 2)}^{2} - 4 x + 8 - 21 = 0$

Subtract $21$ from $8$ .

${(x - 2)}^{2} - 4 x - 13 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 8$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4$ by $- 9$ .

$x = \frac{8 \pm \sqrt{64 + 36}}{2 \cdot 1}$

Add $64$ and $36$ .

$x = \frac{8 \pm \sqrt{100}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{8 \pm \sqrt{10^{2}}}{2 \cdot 1}$

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

$x = \frac{8 \pm 10}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 10}{2}$

Simplify $\frac{8 \pm 10}{2}$ .

$x = 4 \pm 5$

Step 5

The final answer is the combination of both solutions.

$x = 9, - 1$