Solve Using the Quadratic Formula 18=-2(x-1)^2

$18 = - 2 {(x - 1)}^{2}$

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

Simplify the right side.

Simplify $- 2 {(x - 1)}^{2}$ .

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$18 = - 2 ((x - 1) (x - 1))$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$18 = - 2 (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$18 = - 2 (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$18 = - 2 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$18 = - 2 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Move $- 1$ to the left of $x$ .

$18 = - 2 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$18 = - 2 (x^{2} - x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$18 = - 2 (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$18 = - 2 (x^{2} - x - x + 1)$

Subtract $x$ from $- x$ .

$18 = - 2 (x^{2} - 2 x + 1)$

Apply the distributive property.

$18 = - 2 x^{2} - 2 (- 2 x) - 2 \cdot 1$

Simplify.

Multiply $- 2$ by $- 2$ .

$18 = - 2 x^{2} + 4 x - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$18 = - 2 x^{2} + 4 x - 2$

Move all the expressions to the left side of the equation.

Add $2 x^{2}$ to both sides of the equation.

$18 + 2 x^{2} = 4 x - 2$

Subtract $4 x$ from both sides of the equation.

$18 + 2 x^{2} - 4 x = - 2$

Add $2$ to both sides of the equation.

$18 + 2 x^{2} - 4 x + 2 = 0$

Add $18$ and $2$ .

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 4$ , and $c = 20$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8 \cdot 20}}{2 \cdot 2}$

Multiply $- 8$ by $20$ .

$x = \frac{4 \pm \sqrt{16 - 160}}{2 \cdot 2}$

Subtract $160$ from $16$ .

$x = \frac{4 \pm \sqrt{- 144}}{2 \cdot 2}$

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 2}$

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

$x = \frac{4 \pm i \cdot \sqrt{144}}{2 \cdot 2}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 2}$

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

$x = \frac{4 \pm i \cdot 12}{2 \cdot 2}$

Move $12$ to the left of $i$ .

$x = \frac{4 \pm 12 i}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{4 \pm 12 i}{4}$

Simplify $\frac{4 \pm 12 i}{4}$ .

$x = 1 \pm 3 i$

Solve Using the Quadratic Formula 18=-2(x-1)^2

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