To solve the given problem, we need to compute the expression [tex]\((2+3i)^2 + (2-3i)^2\)[/tex] and determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] when the result is written in the form [tex]\(a + bi\)[/tex].
1. First, let's compute [tex]\((2+3i)^2\)[/tex]:
[tex]\[
(2+3i)^2 = (2+3i)(2+3i)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
(2+3i)(2+3i) = 2 \cdot 2 + 2 \cdot 3i + 3i \cdot 2 + 3i \cdot 3i
\][/tex]
[tex]\[
= 4 + 6i + 6i + 9i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 4 + 12i + 9(-1)
\][/tex]
[tex]\[
= 4 + 12i - 9
\][/tex]
[tex]\[
= -5 + 12i
\][/tex]
2. Next, let's compute [tex]\((2-3i)^2\)[/tex]:
[tex]\[
(2-3i)^2 = (2-3i)(2-3i)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
(2-3i)(2-3i) = 2 \cdot 2 + 2 \cdot (-3i) + (-3i) \cdot 2 + (-3i) \cdot (-3i)
\][/tex]
[tex]\[
= 4 - 6i - 6i + 9i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 4 - 12i + 9(-1)
\][/tex]
[tex]\[
= 4 - 12i - 9
\][/tex]
[tex]\[
= -5 - 12i
\][/tex]
3. Now, add the two results together:
[tex]\[
(2+3i)^2 + (2-3i)^2 = (-5 + 12i) + (-5 - 12i)
\][/tex]
[tex]\[
= -5 + 12i - 5 - 12i
\][/tex]
[tex]\[
= -10 + 0i
\][/tex]
4. Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -10
\][/tex]
[tex]\[
b = 0
\][/tex]
So the complete statement is:
If [tex]\((2+3i)^2 + (2-3i)^2 = a + bi\)[/tex], [tex]\(a = -10\)[/tex] and [tex]\(b = 0\)[/tex].
