Question

which expressions are equivalent to (-9 + 2i)(3 - 6i)? select all that apply. a -39 + 60i b -27 + 54i + 6i - 12i² c -27 + 54i + 6i - 12i² d -15 + 60i

Explanation:

Step1: Expand the product

\[

$$\begin{align*} (-9 + 2i)(3-6i)&=-9\times3-9\times(-6i)+2i\times3+2i\times(-6i)\\ &=-27 + 54i+6i-12i^{2} \end{align*}$$

\]

Step2: Recall \(i^{2}=-1\)

Since \(i^{2}=-1\), then \(-27 + 54i+6i-12i^{2}=-27 + 54i+6i-12\times(-1)\).

Step3: Simplify the expression

\[

$$\begin{align*} -27 + 54i+6i + 12&=(-27 + 12)+(54i+6i)\\ &=-15+60i \end{align*}$$

\]
Also, we can rewrite the steps in a different - order. If we first expand and keep \(i^{2}\) as it is:
\[

$$\begin{align*} (-9 + 2i)(3-6i)&=-27+54i + 6i-12i^{2}\\ &=-27+60i-12i^{2} \end{align*}$$

\]
If we consider the form without fully simplifying the real - part addition:
\[

$$\begin{align*} -27+60i-12\times(-1)&=-27 + 12+60i\\ &=-39+60i \end{align*}$$

\]
So the equivalent expressions are - 15 + 60i and - 39+60i.

Answer:

A. - 39 + 60i, D. - 15+60i