Complex number square roots

Question

I am very confused how to do this problem and I would appreciate if someone could explain. Cheers

Suppose u+vi is a square root of x+yi. Express the following in terms of x and/or y only.

u^2 - v^2 = ?

2uv = ?

u^2 + v^2 = ?

Hence find the square root of 8-15i

Answer 1

As $u+vi$ is a square root of $x+yi$, it follows that $(u+vi)^2 = x+yi$. Expanding, we obtain $u^2+2uvi-v^2 = x+yi$. Assuming that $u,v,x,y \in \mathbb{R}$, equating real parts gives $x = u^2-v^2$, and equating imaginary parts gives $y = 2uv$. We then have \begin{align*}\left(u^2+v^2\right)^2 &= u^4+2u^2v^2+v^4 \\ &= \left(u^4-2u^2v^2+v^4\right) + 4u^2v^2 \\ &= \left(u^2-v^2\right)^2+(2uv)^2 \\ &= x^2+y^2\end{align*} so, as $u^2 + v^2 \geq 0$, we take the positive square root to get $u^2 + v^2 = \sqrt{x^2+y^2}$.

Now, to find the square root of $8-15i$, we can use the above results to get $u^2-v^2 = 8$ and $u^2+v^2=\sqrt{8^2+(-15)^2} = 17$. Thus $2u^2 = 8 + 17 = 25$, so $u = \pm\frac{5}{\sqrt{2}}$. As $2uv = -15$, we deduce $v = \mp\frac{3}{\sqrt{2}}$. This means the square roots of $8+15i$ are \begin{equation*}\frac{5}{\sqrt{2}} - \frac{3}{\sqrt{2}}i \quad\text{ and }\quad -\frac{5}{\sqrt{2}} + \frac{3}{\sqrt{2}}i\end{equation*}

Answer 2

Presumably, $u,v,x,y$ are all real numbers.

Note that "$u+vi$ is a square root of $x + yi$" means that $(u+vi)^2 = x+yi$. Note that $$ (u+vi)^2 = (u+vi)(u+vi) = u^2 - 2uvi + vi^2 = (u^2 - v^2) - (2uv)i. $$ Why does this tell you the answer to the first two questions?

There are a few ways to do the third one, but the easiest approach here is to note that $$ |u + vi|^2 = |(u+vi)^2|, $$ which holds as a consequence of the general fact that $|wz| = |w|\cdot|z|$ for complex numbers $w,z$.