To simplify the expression [tex]\((2 + j s)^2 + \frac{5(7 + j 2)}{3 - j 4} - j (4 - j 6)\)[/tex] and express it in the form [tex]\(a + j b\)[/tex], follow these steps:
### Step-by-Step Solution
1. Simplify [tex]\((2 + j s)^2\)[/tex]:
[tex]\[
(2 + j s)^2 = (2 + j s)(2 + j s)
\][/tex]
Using the distributive property:
[tex]\[
= 2(2) + 2(j s) + (j s)(2) + (j s)(j s)
\][/tex]
Recall that [tex]\(j^2 = -1\)[/tex]:
[tex]\[
= 4 + 2j s + 2j s + j^2 s^2
\][/tex]
[tex]\[
= 4 + 4j s - s^2
\][/tex]
2. Simplify [tex]\(\frac{5(7 + j 2)}{3 - j 4}\)[/tex]:
Let's denote the numerator as [tex]\(5(7 + j 2)\)[/tex] and the denominator as [tex]\(3 - j 4\)[/tex]:
[tex]\[
5(7 + j 2) = 35 + 10j
\][/tex]
To handle the division by a complex number, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{35 + 10j}{3 - j 4} \cdot \frac{3 + j 4}{3 + j 4}
\][/tex]
The denominator becomes:
[tex]\[
(3 - j 4)(3 + j 4) = 9 + 12j - 12j - 16j^2 = 9 + 16 = 25
\][/tex]
And the numerator:
[tex]\[
(35 + 10j)(3 + j 4) = 35 \cdot 3 + 35 \cdot j 4 + 10j \cdot 3 + 10j \cdot j 4
\][/tex]
[tex]\[
= 105 + 140j + 30j - 40 = 65 + 170j
\][/tex]
Combining:
[tex]\[
\frac{65 + 170j}{25} = \frac{65}{25} + j \frac{170}{25} = \frac{13}{5} + j \frac{34}{5}
\][/tex]
3. Simplify [tex]\(-j (4 - j 6)\)[/tex]:
[tex]\[
-j (4 - j 6) = -j \cdot 4 + j^2 \cdot 6
\][/tex]
Recall that [tex]\(j^2 = -1\)[/tex]:
[tex]\[
= -4j + 6(-1) = -4j - 6
\][/tex]
4. Combine all parts:
Now combining all the simplified parts:
[tex]\[
(4 + 4j s - s^2) + \left( \frac{13}{5} + j \frac{34}{5} \right) - (6 + 4j)
\][/tex]
Group the real parts and the imaginary parts together:
[tex]\[
\left(4 + \frac{13}{5} - 6 - s^2\right) + \left(4j s + j \frac{34}{5} - 4j\right)
\][/tex]
Simplify the real and imaginary parts:
[tex]\[
4 - 6 + \frac{13}{5} = -2 + \frac{13}{5} = -2 + 2.6 = 0.6 = \frac{3}{5}
\][/tex]
[tex]\[
4j s + j \frac{34}{5} - 4j = j \left(4 s + \frac{34}{5} - 4\right) = j \left(4 s + \frac{34 - 20}{5}\right) = j \left(4 s + \frac{14}{5}\right)
\][/tex]
5. Write the simplified expression:
Combine these results:
[tex]\[
-s^2 + \frac{3}{5} + j \left(4 s + \frac{14}{5}\right)
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
a + j b = -s^2 + \frac{3}{5} + j \left(4 s + \frac{14}{5}\right)
\][/tex]
Or separating real and imaginary parts clearly:
- Real part: [tex]\( \boxed{-s^2 + \frac{3}{5}} \)[/tex]
- Imaginary part: [tex]\( \boxed{4 s + \frac{14}{5}} \)[/tex]
