Question

Obtain an expression that looks like this: 25+ 4 А B. X(S) = + s(s +6) S +6 S

A= ___________
B =____________

Answer 1

`A = (2)/(3)`

`B = (4)/(3)`

Explanation:

Given

`X(s) = (2s + 4)/(s(s +6))= (A)/(s) + (B)/(s + 6)`

Required

Find A and B

The expression can be rewritten as

`(2s + 4)/(s(s +6))= (A)/(s) + (B)/(s + 6)`

Take LCM of the right hand side

`(2s + 4)/(s(s +6))= (A(s + 6 )+ Bs)/(s(s +6))`

Cancel out the denominators

`2s + 4 = A(s + 6) + Bs`

Open Bracket

`2s + 4 = As + 6A+Bs`

Reorder

`2s + 4 = As +Bs+ 6A`

By direct comparison:

`2s = As + Bs`

`4 = 6A`

Solving
`2s = As + Bs`

Divide through by s

`2 = A + B`

Solve for A in
`4 = 6A`

`(4)/(6) = A`

`A = (4)/(6)`

`A = (2)/(3)`

Substitute
`(2)/(3)` for A in
`2 = A + B`

`2 = (2)/(3) + B`

`B = 2 - (2)/(3)`

LCM

`B = (6 - 2)/(3)`

`B = (4)/(3)`