Question

Perform the indicated operation & simplify. Express the answer in terms of i (as a complex number).
(-12+6i)^2

Answer 1

To perform the operation \((-12 + 6i)^2\), we can apply the formula for squaring a binomial:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

In this case, let \(a = -12\) and \(b = 6i\).

Calculating each term:

1. \(a^2 = (-12)^2 = 144\)
2. \(b^2 = (6i)^2 = 36i^2\). Since \(i^2 = -1\), we have \(36i^2 = 36(-1) = -36\).
3. \(2ab = 2 \cdot (-12) \cdot (6i) = -144i\).

Now, we can sum these terms:

\[
(-12 + 6i)^2 = a^2 + 2ab + b^2 = 144 - 36 - 144i
\]

Combining the real parts:

\[
144 - 36 = 108
\]

So, we have:

\[
(-12 + 6i)^2 = 108 - 144i
\]

Thus, the final result is:

\[
\boxed{108 - 144i}
\]