Factor over the Complex Numbers 12m^2+27

$12 m^{2} + 27$

Step 1

Factor $3$ out of $12 m^{2} + 27$ .

Factor $3$ out of $12 m^{2}$ .

$3 (4 m^{2}) + 27$

Factor $3$ out of $27$ .

$3 (4 m^{2}) + 3 (9)$

Factor $3$ out of $3 (4 m^{2}) + 3 (9)$ .

$3 (4 m^{2} + 9)$

Step 2

Use the quadratic formula to find the roots for $4 m^{2} + 9 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 4$ , $b = 0$ , and $c = 9$ into the quadratic formula and solve for $m$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (4 \cdot 9)}}{2 \cdot 4}$

Simplify.

Simplify the numerator.

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

$m = \frac{0 \pm \sqrt{0 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 9$ .

Multiply $- 4$ by $4$ .

$m = \frac{0 \pm \sqrt{0 - 16 \cdot 9}}{2 \cdot 4}$

Multiply $- 16$ by $9$ .

$m = \frac{0 \pm \sqrt{0 - 144}}{2 \cdot 4}$

Subtract $144$ from $0$ .

$m = \frac{0 \pm \sqrt{- 144}}{2 \cdot 4}$

Rewrite $- 144$ as $- 1 (144)$ .

$m = \frac{0 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$m = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 4}$

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

$m = \frac{0 \pm i \cdot \sqrt{144}}{2 \cdot 4}$

Rewrite $144$ as $12^{2}$ .

$m = \frac{0 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 4}$

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{0 \pm i \cdot 12}{2 \cdot 4}$

Move $12$ to the left of $i$ .

$m = \frac{0 \pm 12 i}{2 \cdot 4}$

Multiply $2$ by $4$ .

$m = \frac{0 \pm 12 i}{8}$

Simplify $\frac{0 \pm 12 i}{8}$ .

$m = \frac{\pm 3 i}{2}$

Step 3

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

$12 (m - \frac{3 i}{2}) (m - (\frac{- 3 i}{2}))$

Step 4

Simplify the factored form.

$12 (m - \frac{3 i}{2}) (m + \frac{3 i}{2})$

Factor over the Complex Numbers 12m^2+27

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