Find All Complex Solutions (x-2)^2=8

${(x - 2)}^{2} = 8$

Step 1

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

Subtract $8$ from both sides of the equation.

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

Simplify ${(x - 2)}^{2} - 8$ .

Simplify each term.

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

$(x - 2) (x - 2) - 8 = 0$

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

Apply the distributive property.

$x (x - 2) - 2 (x - 2) - 8 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) - 8 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 8 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 8 = 0$

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

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 8 = 0$

Multiply $- 2$ by $- 2$ .

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

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

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

Subtract $8$ from $4$ .

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

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 $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 4$ .

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

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 1}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{2}}{2}$

Simplify $\frac{4 \pm 4 \sqrt{2}}{2}$ .

$x = 2 \pm 2 \sqrt{2}$

Step 5

The final answer is the combination of both solutions.

$x = 2 + 2 \sqrt{2}, 2 - 2 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 2 + 2 \sqrt{2}, 2 - 2 \sqrt{2}$

Decimal Form:

$x = 4.82842712 \dots, - 0.82842712 \dots$

Find All Complex Solutions (x-2)^2=8

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