Textbook Problem

Some of the solution sets for quadratic equations in the next sections in this chapter will contain complex numbers such as $\frac{-4+\sqrt{-12}}{2}$ and $\frac{-4-\sqrt{-12}}{2}$. We can simplify the first number as follows.

$\begin{array}{rll}\frac{-4+\sqrt{-12}}{2}& =& \frac{-4+i\sqrt{12}}{2}\\ & =& \frac{-4+2i\sqrt{3}}{2}& =& \frac{2\left(2+i\sqrt{3}\right)}{2}\\ & =& -2+i\sqrt{3}\end{array}$

Simplify each of the following complex numbers.

(Objective 3)

(a) $\frac{-4-\sqrt{-12}}{2}$	(b) $\frac{6+\sqrt{-24}}{4}$
(c) $\frac{-1-\sqrt{-18}}{2}$	(d) $\frac{-6+\sqrt{-27}}{3}$
(e) $\frac{10+\sqrt{-45}}{4}$	(f) $\frac{4-\sqrt{-48}}{2}$