Answer:
[tex]A = \frac{2}{3}[/tex]
[tex]B = \frac{4}{3}[/tex]
Step-by-step explanation:
Given
[tex]X(s) = \frac{2s + 4}{s(s +6)}= \frac{A}{s} + \frac{B}{s + 6}[/tex]
Required
Find A and B
The expression can be rewritten as
[tex]\frac{2s + 4}{s(s +6)}= \frac{A}{s} + \frac{B}{s + 6}[/tex]
Take LCM of the right hand side
[tex]\frac{2s + 4}{s(s +6)}= \frac{A(s + 6 )+ Bs}{s(s +6)}[/tex]
Cancel out the denominators
[tex]2s + 4 = A(s + 6) + Bs[/tex]
Open Bracket
[tex]2s + 4 = As + 6A+Bs[/tex]
Reorder
[tex]2s + 4 = As +Bs+ 6A[/tex]
By direct comparison:
[tex]2s = As + Bs[/tex]
[tex]4 = 6A[/tex]
Solving [tex]2s = As + Bs[/tex]
Divide through by s
[tex]2 = A + B[/tex]
Solve for A in [tex]4 = 6A[/tex]
[tex]\frac{4}{6} = A[/tex]
[tex]A = \frac{4}{6}[/tex]
[tex]A = \frac{2}{3}[/tex]
Substitute [tex]\frac{2}{3}[/tex] for A in [tex]2 = A + B[/tex]
[tex]2 = \frac{2}{3} + B[/tex]
[tex]B = 2 - \frac{2}{3}[/tex]
LCM
[tex]B = \frac{6 - 2}{3}[/tex]
[tex]B = \frac{4}{3}[/tex]
