If [tex]$\frac{2-2i}{3+4i}$[/tex] is equal to [tex]$a+bi$[/tex], what is the value of [tex]$a$[/tex]?

A. [tex]$\frac{-2}{25}$[/tex]
B. [tex]$\frac{-14}{25}$[/tex]
C. [tex]$\frac{-1}{2}$[/tex]
D. [tex]$\frac{2}{3}$[/tex]



Answer :

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]