To solve the given problem, we need to multiply the two complex numbers [tex]\( (3 - 4i) \)[/tex] and [tex]\( (3 + 4i) \)[/tex].
Recall that when multiplying two complex numbers, we use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(a + bi)(c + di) = ac + adi + bci + bdi^2
\][/tex]
We'll apply this method:
1. [tex]\( 3 \times 3 \)[/tex] which gives [tex]\( 9 \)[/tex].
2. [tex]\( 3 \times 4i \)[/tex] which gives [tex]\( 12i \)[/tex].
3. [tex]\( -4i \times 3 \)[/tex] which gives [tex]\( -12i \)[/tex].
4. [tex]\( -4i \times 4i \)[/tex] which gives [tex]\( -16i^2 \)[/tex].
Combine the real and imaginary parts:
[tex]\[
(3 - 4i)(3 + 4i) = 9 + 12i - 12i - 16i^2
\][/tex]
Notice that the imaginary parts [tex]\( 12i \)[/tex] and [tex]\( -12i \)[/tex] cancel each other out:
[tex]\[
9 + 12i - 12i - 16i^2 = 9 - 16i^2
\][/tex]
Recall that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
9 - 16(-1) = 9 + 16
\][/tex]
Adding these together gives:
[tex]\[
25
\][/tex]
So, the product of [tex]\( (3 - 4i) \)[/tex] and [tex]\( (3 + 4i) \)[/tex] is [tex]\( 25 + 0i \)[/tex].
The correct answer is:
[tex]\[ 25 + 0i \][/tex]
