To solve the division [tex]\(\frac{1+2i}{5-i}\)[/tex] and express the result in the form [tex]\(a + bi\)[/tex], follow the steps below:
1. Express the division in terms of real and imaginary parts:
[tex]\[
\frac{1+2i}{5-i}
\][/tex]
2. Multiply the numerator and denominator by the conjugate of the denominator:
The conjugate of [tex]\(5 - i\)[/tex] is [tex]\(5 + i\)[/tex]. To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by [tex]\(5 + i\)[/tex]:
[tex]\[
\frac{(1+2i)(5+i)}{(5-i)(5+i)}
\][/tex]
3. Simplify the denominator:
The denominator becomes:
[tex]\[
(5-i)(5+i) = 5^2 - (-i^2) = 25 + 1 = 26
\][/tex]
4. Expand the numerator using the distributive property:
[tex]\[
(1+2i)(5+i) = 1 \cdot 5 + 1 \cdot i + 2i \cdot 5 + 2i \cdot i = 5 + i + 10i + 2i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
5 + i + 10i + 2(-1) = 5 + 11i - 2 = 3 + 11i
\][/tex]
5. Combine the results:
[tex]\[
\frac{3 + 11i}{26}
\][/tex]
6. Separate the real and imaginary parts:
The fraction can be written as:
[tex]\[
\frac{3}{26} + \frac{11i}{26}
\][/tex]
Thus:
[tex]\[
a = \frac{3}{26} \approx 0.11538461538461538
\][/tex]
[tex]\[
b = \frac{11}{26} \approx 0.4230769230769231
\][/tex]
So the result of the division [tex]\(\frac{1+2i}{5-i}\)[/tex] in the form [tex]\(a + bi\)[/tex] is:
[tex]\[
a \approx 0.11538461538461538
\][/tex]
[tex]\[
b \approx 0.4230769230769231
\][/tex]
