Factor over the Complex Numbers x^2+11

$x^{2} + 11$

Use the quadratic formula to find the roots for $x^{2} + 11 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 0$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 11}}{2 \cdot 1}$

Multiply $- 4$ by $11$ .

$x = \frac{0 \pm \sqrt{0 - 44}}{2 \cdot 1}$

Subtract $44$ from $0$ .

$x = \frac{0 \pm \sqrt{- 44}}{2 \cdot 1}$

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 44}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot 1}$

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

$x = \frac{0 \pm i \cdot \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{0 \pm i \cdot \sqrt{4 (11)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (2 \sqrt{11})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{0 \pm 2 i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{0 \pm 2 i \sqrt{11}}{2}$

Simplify $\frac{0 \pm 2 i \sqrt{11}}{2}$ .

$x = \pm i \sqrt{11}$

Find the factors from the roots, then multiply the factors together.

$(x - (i \sqrt{11})) (x - (- i \sqrt{11}))$

Simplify the factored form.

$(x - i \sqrt{11}) (x + i \sqrt{11})$