Find a and b (both real) when (a+b*i)^2=i.
Every complex number has a real and imaginary part. For the complex number z=a+bi the notation for real and imaginary parts respectively are Re(z)=a and Im(z)=b. If you know this, many complex algebra equations will become much simpler to solve.
In this specific case, firstly consider LHS, giving z=a2+ 2abi+(ib)2=(a2-b2)+(2ab)*i. (since i2=-1). Consequently, Re(z)=a2-b2 and Im(z)=2ab. Next consider the RHS, write its real and imaginary parts: Re(i)=0 and Im(i)=1. Equate LHS and RHS, getting a system of equations: a2-b2=0 and 2ab=1.
The solutions are a=-1/sq(2), b=1/sq(2) and a=1/sq(2), b=-1/sq(2).
