Find All Complex Solutions -11x^2=x+11

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

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

Subtract $x$ from both sides of the equation.

$- 11 x^{2} - x = 11$

Subtract $11$ from both sides of the equation.

$- 11 x^{2} - x - 11 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 11$ , $b = - 1$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 11 \cdot - 11)}}{2 \cdot - 11}$

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 11 \cdot - 11}}{2 \cdot - 11}$

Multiply $- 4$ by $- 11$ .

$x = \frac{1 \pm \sqrt{1 + 44 \cdot - 11}}{2 \cdot - 11}$

Multiply $44$ by $- 11$ .

$x = \frac{1 \pm \sqrt{1 - 484}}{2 \cdot - 11}$

Subtract $484$ from $1$ .

$x = \frac{1 \pm \sqrt{- 483}}{2 \cdot - 11}$

Rewrite $- 483$ as $- 1 (483)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 483}}{2 \cdot - 11}$

Rewrite $\sqrt{- 1 (483)}$ as $\sqrt{- 1} \cdot \sqrt{483}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{483}}{2 \cdot - 11}$

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

$x = \frac{1 \pm i \sqrt{483}}{2 \cdot - 11}$

Multiply $2$ by $- 11$ .

$x = \frac{1 \pm i \sqrt{483}}{- 22}$

Move the negative in front of the fraction.

$x = - \frac{1 \pm i \sqrt{483}}{22}$