Find All Complex Solutions 2(x+5)^2=18

$2 {(x + 5)}^{2} = 18$

Step 1

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

Subtract $18$ from both sides of the equation.

$2 {(x + 5)}^{2} - 18 = 0$

Simplify $2 {(x + 5)}^{2} - 18$ .

Simplify each term.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$2 ((x + 5) (x + 5)) - 18 = 0$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$2 (x (x + 5) + 5 (x + 5)) - 18 = 0$

Apply the distributive property.

$2 (x \cdot x + x \cdot 5 + 5 (x + 5)) - 18 = 0$

Apply the distributive property.

$2 (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 18 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 18 = 0$

Move $5$ to the left of $x$ .

$2 (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) - 18 = 0$

Multiply $5$ by $5$ .

$2 (x^{2} + 5 x + 5 x + 25) - 18 = 0$

Add $5 x$ and $5 x$ .

$2 (x^{2} + 10 x + 25) - 18 = 0$

Apply the distributive property.

$2 x^{2} + 2 (10 x) + 2 \cdot 25 - 18 = 0$

Simplify.

Multiply $10$ by $2$ .

$2 x^{2} + 20 x + 2 \cdot 25 - 18 = 0$

Multiply $2$ by $25$ .

$2 x^{2} + 20 x + 50 - 18 = 0$

Subtract $18$ from $50$ .

$2 x^{2} + 20 x + 32 = 0$

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 $a = 2$ , $b = 20$ , and $c = 32$ into the quadratic formula and solve for $x$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (2 \cdot 32)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $20$ to the power of $2$ .

$x = \frac{- 20 \pm \sqrt{400 - 4 \cdot 2 \cdot 32}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 32$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 20 \pm \sqrt{400 - 8 \cdot 32}}{2 \cdot 2}$

Multiply $- 8$ by $32$ .

$x = \frac{- 20 \pm \sqrt{400 - 256}}{2 \cdot 2}$

Subtract $256$ from $400$ .

$x = \frac{- 20 \pm \sqrt{144}}{2 \cdot 2}$

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

$x = \frac{- 20 \pm \sqrt{12^{2}}}{2 \cdot 2}$

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

$x = \frac{- 20 \pm 12}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 20 \pm 12}{4}$

Simplify $\frac{- 20 \pm 12}{4}$ .

$x = - 5 \pm 3$

Step 5

The final answer is the combination of both solutions.

$x = - 2, - 8$

Find All Complex Solutions 2(x+5)^2=18

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