To solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the complex number division [tex]\(\frac{2 - 2i}{3 + 4i}\)[/tex], we'll follow these steps:
1. Find 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 to remove the imaginary part from the denominator:
[tex]\[
\frac{2 - 2i}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i}
\][/tex]
3. Perform the multiplication in the numerator and the denominator:
[tex]\[
\text{Numerator: } (2 - 2i)(3 - 4i)
\][/tex]
Applying the distributive property (FOIL method):
[tex]\[
= 2 \cdot 3 + 2 \cdot (-4i) - 2i \cdot 3 - 2i \cdot (-4i)
\][/tex]
[tex]\[
= 6 - 8i - 6i + 8i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 6 - 8i - 6i + 8(-1)
\][/tex]
[tex]\[
= 6 - 8i - 6i - 8
\][/tex]
[tex]\[
= -2 - 14i
\][/tex]
Now for the denominator:
[tex]\[
\text{Denominator: } (3 + 4i)(3 - 4i)
\][/tex]
Applying the distributive property:
[tex]\[
= 3 \cdot 3 + 3 \cdot (-4i) + 4i \cdot 3 + 4i \cdot (-4i)
\][/tex]
[tex]\[
= 9 - 12i + 12i - 16i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 9 - 12i + 12i + 16
\][/tex]
[tex]\[
= 9 + 16
\][/tex]
[tex]\[
= 25
\][/tex]
4. Combine the results to form the final division result:
[tex]\[
\frac{-2 - 14i}{25}
\][/tex]
[tex]\[
= \frac{-2}{25} + \frac{-14i}{25}
\][/tex]
This gives us [tex]\(-\frac{2}{25} + -\frac{14}{25}i\)[/tex].
From this, we clearly see that [tex]\(a = -\frac{2}{25}\)[/tex].
Therefore, the value of [tex]\(a\)[/tex] is:
A) [tex]\(\frac{-2}{25}\)[/tex]
