Solve by Completing the Square x^2+23x+111=11x

$x^{2} + 23 x + 111 = 11 x$

Step 1

Simplify the equation into a proper form to complete the square.

Subtract $111$ from both sides of the equation.

$x^{2} + 23 x = 11 x - 111$

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

Subtract $11 x$ from both sides of the equation.

$x^{2} + 23 x - 11 x = - 111$

Subtract $11 x$ from $23 x$ .

$x^{2} + 12 x = - 111$

Step 2

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

${(\frac{b}{2})}^{2} = {(6)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 12 x + {(6)}^{2} = - 111 + {(6)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $6$ to the power of $2$ .

$x^{2} + 12 x + 36 = - 111 + {(6)}^{2}$

Simplify the right side.

Simplify $- 111 + {(6)}^{2}$ .

Raise $6$ to the power of $2$ .

$x^{2} + 12 x + 36 = - 111 + 36$

Add $- 111$ and $36$ .

$x^{2} + 12 x + 36 = - 75$

Step 5

Factor the perfect trinomial square into ${(x + 6)}^{2}$ .

${(x + 6)}^{2} = - 75$

Step 6

Solve the equation for $x$ .

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

$x + 6 = \pm \sqrt{- 75}$

Simplify $\pm \sqrt{- 75}$ .

Rewrite $- 75$ as $- 1 (75)$ .

$x + 6 = \pm \sqrt{- 1 (75)}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x + 6 = \pm \sqrt{- 1} \cdot \sqrt{75}$

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

$x + 6 = \pm i \cdot \sqrt{75}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$x + 6 = \pm i \cdot \sqrt{25 (3)}$

Rewrite $25$ as $5^{2}$ .

$x + 6 = \pm i \cdot \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$x + 6 = \pm i \cdot (5 \sqrt{3})$

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

$x + 6 = \pm 5 i \sqrt{3}$

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

Subtract $6$ from both sides of the equation.

$x = \pm 5 i \sqrt{3} - 6$

Reorder $\pm 5 i \sqrt{3}$ and $- 6$ .

$x = - 6 \pm 5 i \sqrt{3}$

Solve by Completing the Square x^2+23x+111=11x

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