Solve by Completing the Square 2x^2+9x=x-368

$2 x^{2} + 9 x = x - 368$

Step 1

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$2 x^{2} + 9 x - x = - 368$

Subtract $x$ from $9 x$ .

$2 x^{2} + 8 x = - 368$

Step 2

Divide each term in $2 x^{2} + 8 x = - 368$ by $2$ and simplify.

Divide each term in $2 x^{2} + 8 x = - 368$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{8 x}{2} = \frac{- 368}{2}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{8 x}{2} = \frac{- 368}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{8 x}{2} = \frac{- 368}{2}$

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8 x$ .

$x^{2} + \frac{2 (4 x)}{2} = \frac{- 368}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (4 x)}{2 (1)} = \frac{- 368}{2}$

Cancel the common factor.

$x^{2} + \frac{2 (4 x)}{2 \cdot 1} = \frac{- 368}{2}$

Rewrite the expression.

$x^{2} + \frac{4 x}{1} = \frac{- 368}{2}$

Divide $4 x$ by $1$ .

$x^{2} + 4 x = \frac{- 368}{2}$

Simplify the right side.

Divide $- 368$ by $2$ .

$x^{2} + 4 x = - 184$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(2)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + 4 x + {(2)}^{2} = - 184 + {(2)}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

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

$x^{2} + 4 x + 4 = - 184 + {(2)}^{2}$

Simplify the right side.

Simplify $- 184 + {(2)}^{2}$ .

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

$x^{2} + 4 x + 4 = - 184 + 4$

Add $- 184$ and $4$ .

$x^{2} + 4 x + 4 = - 180$

Step 6

Factor the perfect trinomial square into ${(x + 2)}^{2}$ .

${(x + 2)}^{2} = - 180$

Step 7

Solve the equation for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x + 2 = \pm \sqrt{- 180}$

Simplify $\pm \sqrt{- 180}$ .

Rewrite $- 180$ as $- 1 (180)$ .

$x + 2 = \pm \sqrt{- 1 (180)}$

Rewrite $\sqrt{- 1 (180)}$ as $\sqrt{- 1} \cdot \sqrt{180}$ .

$x + 2 = \pm \sqrt{- 1} \cdot \sqrt{180}$

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

$x + 2 = \pm i \cdot \sqrt{180}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$x + 2 = \pm i \cdot \sqrt{36 (5)}$

Rewrite $36$ as $6^{2}$ .

$x + 2 = \pm i \cdot \sqrt{6^{2} \cdot 5}$

Pull terms out from under the radical.

$x + 2 = \pm i \cdot (6 \sqrt{5})$

Move $6$ to the left of $i$ .

$x + 2 = \pm 6 i \sqrt{5}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$x = \pm 6 i \sqrt{5} - 2$

Reorder $\pm 6 i \sqrt{5}$ and $- 2$ .

$x = - 2 \pm 6 i \sqrt{5}$

Solve by Completing the Square 2x^2+9x=x-368

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