Question

practice assignment 5.1 quadratic functions

question 7

simplify the radical expression \frac{-12 - \sqrt{-320}}{-8}.

write your answer in exact form without decimals. to enter an expression such as \frac{8 + 7i\sqrt{13}}{3} type (8+7isqrt(13))/3.

the result is

Explanation:

⚡ Using what you learned: simplifying radical expressions · Introduction to Complex Numbers

Step 1: Simplify the negative radical

First, rewrite the square root of the negative number using the imaginary unit \( i \), where \( \sqrt{-1} = i \):

$$ \sqrt{-320} = \sqrt{-1 \cdot 320} = i\sqrt{320} $$

Next, find the largest perfect square factor of \( 320 \):

$$ 320 = 64 \cdot 5 $$

Simplify the radical:

$$ i\sqrt{320} = i\sqrt{64 \cdot 5} = 8i\sqrt{5} $$

Step 2: Substitute back and simplify the fraction

Substitute \( 8i\sqrt{5} \) back into the original expression:

$$ \frac{-12 - 8i\sqrt{5}}{-8} $$

Divide each term in the numerator by the denominator \( -8 \):

$$ \frac{-12}{-8} - \frac{8i\sqrt{5}}{-8} = \frac{3}{2} + i\sqrt{5} $$

To write it as a single fraction with a common denominator:

$$ \frac{3 + 2i\sqrt{5}}{2} $$

Answer:

(3+2isqrt(5))/2