Solve by Completing the Square 2x^2-20x+89=-87

$2 x^{2} - 20 x + 89 = - 87$

Step 1

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

Subtract $89$ from both sides of the equation.

$2 x^{2} - 20 x = - 87 - 89$

Subtract $89$ from $- 87$ .

$2 x^{2} - 20 x = - 176$

Step 2

Divide each term in $2 x^{2} - 20 x = - 176$ by $2$ and simplify.

Divide each term in $2 x^{2} - 20 x = - 176$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{- 20 x}{2} = \frac{- 176}{2}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{- 20 x}{2} = \frac{- 176}{2}$

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

$x^{2} + \frac{- 20 x}{2} = \frac{- 176}{2}$

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20 x$ .

$x^{2} + \frac{2 (- 10 x)}{2} = \frac{- 176}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (- 10 x)}{2 (1)} = \frac{- 176}{2}$

Cancel the common factor.

$x^{2} + \frac{2 (- 10 x)}{2 \cdot 1} = \frac{- 176}{2}$

Rewrite the expression.

$x^{2} + \frac{- 10 x}{1} = \frac{- 176}{2}$

Divide $- 10 x$ by $1$ .

$x^{2} - 10 x = \frac{- 176}{2}$

Simplify the right side.

Divide $- 176$ by $2$ .

$x^{2} - 10 x = - 88$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 5)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - 10 x + {(- 5)}^{2} = - 88 + {(- 5)}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Raise $- 5$ to the power of $2$ .

$x^{2} - 10 x + 25 = - 88 + {(- 5)}^{2}$

Simplify the right side.

Simplify $- 88 + {(- 5)}^{2}$ .

Raise $- 5$ to the power of $2$ .

$x^{2} - 10 x + 25 = - 88 + 25$

Add $- 88$ and $25$ .

$x^{2} - 10 x + 25 = - 63$

Step 6

Factor the perfect trinomial square into ${(x - 5)}^{2}$ .

${(x - 5)}^{2} = - 63$

Step 7

Solve the equation for $x$ .

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

$x - 5 = \pm \sqrt{- 63}$

Simplify $\pm \sqrt{- 63}$ .

Rewrite $- 63$ as $- 1 (63)$ .

$x - 5 = \pm \sqrt{- 1 (63)}$

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x - 5 = \pm \sqrt{- 1} \cdot \sqrt{63}$

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

$x - 5 = \pm i \cdot \sqrt{63}$

Rewrite $63$ as $3^{2} \cdot 7$ .

Factor $9$ out of $63$ .

$x - 5 = \pm i \cdot \sqrt{9 (7)}$

Rewrite $9$ as $3^{2}$ .

$x - 5 = \pm i \cdot \sqrt{3^{2} \cdot 7}$

Pull terms out from under the radical.

$x - 5 = \pm i \cdot (3 \sqrt{7})$

Move $3$ to the left of $i$ .

$x - 5 = \pm 3 i \sqrt{7}$

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

Add $5$ to both sides of the equation.

$x = \pm 3 i \sqrt{7} + 5$

Reorder $\pm 3 i \sqrt{7}$ and $5$ .

$x = 5 \pm 3 i \sqrt{7}$

Solve by Completing the Square 2x^2-20x+89=-87

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