Simplify (-20+ square root of -75)/40

$\frac{- 20 + \sqrt{- 75}}{40}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 75$ as $- 1 (75)$ .

$\frac{- 20 + \sqrt{- 1 (75)}}{40}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$\frac{- 20 + \sqrt{- 1} \cdot \sqrt{75}}{40}$

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

$\frac{- 20 + i \cdot \sqrt{75}}{40}$

Step 2

Simplify the numerator.

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$\frac{- 20 + i \cdot \sqrt{25 (3)}}{40}$

Rewrite $25$ as $5^{2}$ .

$\frac{- 20 + i \cdot \sqrt{5^{2} \cdot 3}}{40}$

Pull terms out from under the radical.

$\frac{- 20 + i \cdot (5 \sqrt{3})}{40}$

Move $5$ to the left of $i$ .

$\frac{- 20 + 5 i \sqrt{3}}{40}$

Step 3

Simplify terms.

Cancel the common factor of $- 20 + 5 i \sqrt{3}$ and $40$ .

Factor $5$ out of $- 20$ .

$\frac{5 \cdot - 4 + 5 i \sqrt{3}}{40}$

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

$\frac{5 \cdot - 4 + 5 (i \sqrt{3})}{40}$

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

$\frac{5 \cdot (- 4 + i \sqrt{3})}{40}$

Cancel the common factors.

Factor $5$ out of $40$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 4 + i \sqrt{3}}{8}$

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

$\frac{- 1 (4) + i \sqrt{3}}{8}$

Factor $- 1$ out of $i \sqrt{3}$ .

$\frac{- 1 (4) - (- i \sqrt{3})}{8}$

Factor $- 1$ out of $- 1 (4) - (- i \sqrt{3})$ .

$\frac{- 1 (4 - i \sqrt{3})}{8}$

Move the negative in front of the fraction.

$- \frac{4 - i \sqrt{3}}{8}$