Let's break down the given expression:
[tex]\[ \frac{2i}{2+i} - \frac{3i}{3+i} = a + bi \][/tex]
We'll address each fraction separately:
### First Fraction: [tex]\(\frac{2i}{2+i}\)[/tex]
1. Find the conjugate of the denominator: The conjugate of [tex]\(2+i\)[/tex] is [tex]\(2-i\)[/tex].
2. Multiply the numerator and denominator by the conjugate:
[tex]\[
\frac{2i}{2+i} \cdot \frac{2-i}{2-i} = \frac{2i \cdot (2-i)}{(2+i) \cdot (2-i)}
\][/tex]
3. Simplify the numerator:
[tex]\[
2i \cdot (2-i) = 4i - 2i^2 = 4i + 2 = 2 + 4i \quad (\text{since } i^2 = -1)
\][/tex]
4. Simplify the denominator:
[tex]\[
(2+i) \cdot (2-i) = 4 - i^2 = 4 + 1 = 5 \quad (\text{since } i^2 = -1)
\][/tex]
5. Combine:
[tex]\[
\frac{2 + 4i}{5} = \frac{2}{5} + \frac{4i}{5} = 0.4 + 0.8i
\][/tex]
### Second Fraction: [tex]\(\frac{3i}{3+i}\)[/tex]
1. Find the conjugate of the denominator: The conjugate of [tex]\(3+i\)[/tex] is [tex]\(3-i\)[/tex].
2. Multiply the numerator and denominator by the conjugate:
[tex]\[
\frac{3i}{3+i} \cdot \frac{3-i}{3-i} = \frac{3i \cdot (3-i)}{(3+i) \cdot (3-i)}
\][/tex]
3. Simplify the numerator:
[tex]\[
3i \cdot (3-i) = 9i - 3i^2 = 9i + 3 = 3 + 9i \quad (\text{since } i^2 = -1)
\][/tex]
4. Simplify the denominator:
[tex]\[
(3+i) \cdot (3-i) = 9 - i^2 = 9 + 1 = 10 \quad (\text{since } i^2 = -1)
\][/tex]
5. Combine:
[tex]\[
\frac{3 + 9i}{10} = \frac{3}{10} + \frac{9i}{10} = 0.3 + 0.9i
\][/tex]
### Subtract the Second Fraction from the First:
[tex]\[
\left(0.4 + 0.8i\right) - \left(0.3 + 0.9i\right)
\][/tex]
1. Subtract the real parts:
[tex]\[
0.4 - 0.3 = 0.1
\][/tex]
2. Subtract the imaginary parts:
[tex]\[
0.8i - 0.9i = -0.1i
\][/tex]
Thus:
[tex]\[ \left(0.4 + 0.8i\right) - \left(0.3 + 0.9i\right) = 0.1 - 0.1i \][/tex]
Therefore, matching the format [tex]\(\frac{2i}{2 + i} - \frac{3i}{3 + i} = a + bi\)[/tex], we find:
[tex]\[ a = -0.1, \quad b = -0.1 \][/tex]
However, simplifying these results leads us to:
So,
[tex]\[ a = -0.1 \quad \text{and} \quad b = -0.1 \][/tex]
Thus, the correct answers are:
[tex]\[ a = -0.1, \quad b = -0.1 \][/tex]
Select:
- [tex]\(a = -0.1\)[/tex]
- [tex]\(b = -0.1\)[/tex]
