Simplify (-12+ square root of -45)/24

$\frac{- 12 + \sqrt{- 45}}{24}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 45$ as $- 1 (45)$ .

$\frac{- 12 + \sqrt{- 1 (45)}}{24}$

Rewrite $\sqrt{- 1 (45)}$ as $\sqrt{- 1} \cdot \sqrt{45}$ .

$\frac{- 12 + \sqrt{- 1} \cdot \sqrt{45}}{24}$

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

$\frac{- 12 + i \cdot \sqrt{45}}{24}$

Step 2

Simplify the numerator.

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$\frac{- 12 + i \cdot \sqrt{9 (5)}}{24}$

Rewrite $9$ as $3^{2}$ .

$\frac{- 12 + i \cdot \sqrt{3^{2} \cdot 5}}{24}$

Pull terms out from under the radical.

$\frac{- 12 + i \cdot (3 \sqrt{5})}{24}$

Move $3$ to the left of $i$ .

$\frac{- 12 + 3 i \sqrt{5}}{24}$

Step 3

Simplify terms.

Cancel the common factor of $- 12 + 3 i \sqrt{5}$ and $24$ .

Factor $3$ out of $- 12$ .

$\frac{3 \cdot - 4 + 3 i \sqrt{5}}{24}$

Factor $3$ out of $3 i \sqrt{5}$ .

$\frac{3 \cdot - 4 + 3 (i \sqrt{5})}{24}$

Factor $3$ out of $3 \cdot - 4 + 3 (i \sqrt{5})$ .

$\frac{3 \cdot (- 4 + i \sqrt{5})}{24}$

Cancel the common factors.

Factor $3$ out of $24$ .

$\frac{3 \cdot (- 4 + i \sqrt{5})}{3 (8)}$

Cancel the common factor.

$\frac{3 \cdot (- 4 + i \sqrt{5})}{3 \cdot 8}$

Rewrite the expression.

$\frac{- 4 + i \sqrt{5}}{8}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) + i \sqrt{5}}{8}$

Factor $- 1$ out of $i \sqrt{5}$ .

$\frac{- 1 (4) - (- i \sqrt{5})}{8}$

Factor $- 1$ out of $- 1 (4) - (- i \sqrt{5})$ .

$\frac{- 1 (4 - i \sqrt{5})}{8}$

Move the negative in front of the fraction.

$- \frac{4 - i \sqrt{5}}{8}$