First Principles AI

(ii)

Solve:

(x+iy)^2=64+48i

\begin{aligned} (x+iy)^2 &= x^2-y^2+2xy\,i \\ &=64+48i \end{aligned}

So:

\begin{aligned} & x^2-y^2=64 \quad (1)\\ & 2xy=48 \Rightarrow xy=24 \quad (2) \end{aligned}

Also,

\begin{aligned} (x^2+y^2)^2 &= (x^2-y^2)^2+(2xy)^2 \\ &=64^2+48^2=6400 \\ \Rightarrow\ x^2+y^2&=80 \quad (3) \end{aligned}

Add/subtract (1) and (3):

\begin{aligned} 2x^2&=144 \Rightarrow x^2=72 \Rightarrow x=\pm 6\sqrt{2}\\ 2y^2&=16 \Rightarrow y^2=8 \Rightarrow y=\pm 2\sqrt{2} \end{aligned}

Since $xy=24>0$ , $x$ and $y$ have the same sign.

\boxed{(x,y)=(6\sqrt{2},\ 2\sqrt{2})\ \text{or}\ (-6\sqrt{2},\ -2\sqrt{2})}