Question
Multiply.
(4plus9i)squared
(4plus9i)squared
Answers
Answered by
GPT-4o mini
To multiply \((4 + 9i)^2\), we can use the formula for squaring a binomial, which is \((a + b)^2 = a^2 + 2ab + b^2\). In this case, \(a = 4\) and \(b = 9i\).
Now, let's calculate it:
\[
(4 + 9i)^2 = 4^2 + 2(4)(9i) + (9i)^2
\]
Calculating each term:
1. \(4^2 = 16\)
2. \(2(4)(9i) = 72i\)
3. \((9i)^2 = 81i^2\)
Since \(i^2 = -1\), we have:
\[
81i^2 = 81(-1) = -81
\]
Now, combine all the terms together:
\[
(4 + 9i)^2 = 16 + 72i - 81
\]
Combining the real parts:
\[
16 - 81 = -65
\]
So, the final result is:
\[
-65 + 72i
\]
Thus, \((4 + 9i)^2 = -65 + 72i\).
Now, let's calculate it:
\[
(4 + 9i)^2 = 4^2 + 2(4)(9i) + (9i)^2
\]
Calculating each term:
1. \(4^2 = 16\)
2. \(2(4)(9i) = 72i\)
3. \((9i)^2 = 81i^2\)
Since \(i^2 = -1\), we have:
\[
81i^2 = 81(-1) = -81
\]
Now, combine all the terms together:
\[
(4 + 9i)^2 = 16 + 72i - 81
\]
Combining the real parts:
\[
16 - 81 = -65
\]
So, the final result is:
\[
-65 + 72i
\]
Thus, \((4 + 9i)^2 = -65 + 72i\).