To solve the expression [tex]\(\frac{8 - i}{3 - 2i}\)[/tex] in the form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers, we need to simplify it. Here is the step-by-step procedure to rewrite it and identify the value of [tex]\(a\)[/tex]:
1. Complex Conjugate Multiplication:
Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 - 2i\)[/tex] is [tex]\(3 + 2i\)[/tex].
Thus, the expression becomes:
[tex]\[
\frac{(8 - i)(3 + 2i)}{(3 - 2i)(3 + 2i)}
\][/tex]
2. Simplify the Denominator:
The denominator is a difference of squares which simplifies as follows:
[tex]\[
(3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13
\][/tex]
3. Expand the Numerator:
Next, expand the numerator:
[tex]\[
(8 - i)(3 + 2i) = 8 \cdot 3 + 8 \cdot 2i - i \cdot 3 - i \cdot 2i
\][/tex]
Simplify each term:
[tex]\[
= 24 + 16i - 3i - 2i^2 = 24 + 13i - 2(-1) = 24 + 13i + 2 = 26 + 13i
\][/tex]
4. Divide Terms in the Numerator by the Denominator:
Now, divide each part of the complex number by the denominator 13:
[tex]\[
\frac{26 + 13i}{13} = \frac{26}{13} + \frac{13i}{13} = 2 + i
\][/tex]
5. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
The resulting expression is [tex]\(2 + i\)[/tex]. Here we can see the real part [tex]\(a\)[/tex] is [tex]\(2\)[/tex] and the imaginary part [tex]\(b\)[/tex] is [tex]\(1\)[/tex].
Thus, in the expression [tex]\(\frac{8 - i}{3 - 2i} = 2 + i\)[/tex], the value of [tex]\(a\)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
