Theory Complex multiplication: (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di)=(ac-bd)+(ad+bc)i(a+bi)(c+di)=(ac−bd)+(ad+bc)i. Use i2=−1i^2=-1i2=−1. To solve for x,yx,yx,y in x+iy=α+iβx+iy=\alpha+i\betax+iy=α+iβ, compare real and imaginary parts. To simplify 1a+bi\dfrac{1}{a+bi}a+bi1, multiply numerator and denominator by a−bia-bia−bi.