If the expression below is rewritten in the form [tex]$a+bi$[/tex], where [tex]$a$[/tex] and [tex][tex]$b$[/tex][/tex] are real numbers, what is the value of [tex]$a$[/tex]? (Note: [tex]$i=\sqrt{-1}$[/tex])

[tex]\frac{8-i}{3-2i}[/tex]



Answer :

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]