I need help doing partial fraction decomposition decompose t

I need help doing partial fraction decomposition.

decompose this: 1/(-w^2+7i+10)=A/(iw+2)+B/(iw+5)

all steps explained would really help

i means imaginary number

Solution

There appears to be an error in the expression on the LHS. It is possibly 1/(-w² +7iw+10). If so, we have 1/(-w² +7iw+10) = 1/[ (iw)² +5iw +2iw +10] = 1/ [iw(iw+5) +2(iw+5)] = 1/(iw+5)(iw+2). Then, we have 1/(iw+5)(iw+2) = A/(iw+2)+B/(iw+5) = [A(iw+5)+B(iw+2)]/( iw+5)(iw+2). Now, on multiplyoing both the sides by (iw+5)(iw+2), we get [A(iw+5)+B(iw+2)]=1 or, (5A+2B)+i(A+B) = 1 Now, if A and B are real numbers, we have 5A+2B =1 and A+B = 0 so that B = -A and hence 5A -2A = 1 or, 3A =1 or, A = 1/3 . Then B = -A= -1/3.