To solve the problem [tex]\(\frac{5+6i}{3-4i}\)[/tex], we need to simplify the given complex fraction. Here is a step-by-step solution:
1. Identify the given complex number and its conjugate:
We have the complex number [tex]\(\frac{5 + 6i}{3 - 4i}\)[/tex]. To simplify this, we will multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of [tex]\(3 - 4i\)[/tex] is [tex]\(3 + 4i\)[/tex].
2. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{5 + 6i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i}.
\][/tex]
3. Perform the multiplication in the numerator:
Using the distributive property (FOIL):
[tex]\[
(5 + 6i)(3 + 4i) = 5 \cdot 3 + 5 \cdot 4i + 6i \cdot 3 + 6i \cdot 4i.
\][/tex]
Simplify the terms:
[tex]\[
= 15 + 20i + 18i + 24i^2.
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 15 + 20i + 18i + 24(-1).
\][/tex]
[tex]\[
= 15 + 38i - 24.
\][/tex]
[tex]\[
= -9 + 38i.
\][/tex]
4. Perform the multiplication in the denominator:
[tex]\[
(3 - 4i)(3 + 4i) = 3 \cdot 3 + 3 \cdot 4i - 4i \cdot 3 - 4i \cdot 4i.
\][/tex]
Simplify the terms:
[tex]\[
= 9 + 12i - 12i - 16i^2.
\][/tex]
[tex]\[
= 9 + 0 - 16(-1).
\][/tex]
[tex]\[
= 9 + 16.
\][/tex]
[tex]\[
= 25.
\][/tex]
5. Combine the results:
Now, we have:
[tex]\[
\frac{-9 + 38i}{25}.
\][/tex]
Separate the real and imaginary parts:
[tex]\[
= -\frac{9}{25} + \frac{38}{25}i.
\][/tex]
6. Match with the provided options:
The simplified form matches the option [tex]\(-\frac{9}{25} + \frac{38}{25}i\)[/tex].
So, the correct answer is:
[tex]\[
-\frac{9}{25} + \frac{38}{25}i
\][/tex]
