Find All Complex Solutions (x-2)^2=-16

${(x - 2)}^{2} = - 16$

Step 1

Move all terms to the left side of the equation and simplify.

Add $16$ to both sides of the equation.

${(x - 2)}^{2} + 16 = 0$

Simplify ${(x - 2)}^{2} + 16$ .

Simplify each term.

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(x - 2) (x - 2) + 16 = 0$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$x (x - 2) - 2 (x - 2) + 16 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) + 16 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + 16 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 + 16 = 0$

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 + 16 = 0$

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4 + 16 = 0$

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 4 x + 4 + 16 = 0$

Add $4$ and $16$ .

$x^{2} - 4 x + 20 = 0$

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 $a = 1$ , $b = - 4$ , and $c = 20$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 20)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 20$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 20}}{2 \cdot 1}$

Multiply $- 4$ by $20$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot 1}$

Subtract $80$ from $16$ .

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

Rewrite $- 64$ as $- 1 (64)$ .

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

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot 1}$

Move $8$ to the left of $i$ .

$x = \frac{4 \pm 8 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 8 i}{2}$

Simplify $\frac{4 \pm 8 i}{2}$ .

$x = 2 \pm 4 i$

Find All Complex Solutions (x-2)^2=-16

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