Factor over the Complex Numbers x^2+20

$x^{2} + 20$

Step 1

Use the quadratic formula to find the roots for $x^{2} + 20 = 0$

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 $c = 20$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot 20)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 20$ .

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 20}}{2 \cdot 1}$

Multiply $- 4$ by $20$ .

$x = \frac{0 \pm \sqrt{0 - 80}}{2 \cdot 1}$

Subtract $80$ from $0$ .

$x = \frac{0 \pm \sqrt{- 80}}{2 \cdot 1}$

Rewrite $- 80$ as $- 1 (80)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 1}$

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

$x = \frac{0 \pm i \cdot \sqrt{80}}{2 \cdot 1}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$x = \frac{0 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (4 \sqrt{5})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{0 \pm 4 i \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{0 \pm 4 i \sqrt{5}}{2}$

Simplify $\frac{0 \pm 4 i \sqrt{5}}{2}$ .

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

Step 2

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

$(x - (2 i \sqrt{5})) (x - (- 2 i \sqrt{5}))$

Step 3

Simplify the factored form.

$(x - 2 i \sqrt{5}) (x + 2 i \sqrt{5})$