Find the square root of complex number 3 + 4i
Strategy: write down an equation satisfied by the square root, and solve it algebraically. Method: square root x+iy satisfies (x+iy)2 = 3 + 4i. Expand: x2-y2 +2xyi = 3+4i. Comparing coefficients gives: x2-y2 =3 and 2xy =4. Then substitute y: x2 -4/x2 = 3. Rearrange to get quadratic in x2 : (x2 +1)(x2 -4) = 0. x can't be imaginary (by definition) so x= +/- 2. Plug in to equation 2xy = 4, get y = +/- 1. So square root is +/- (2+i).
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