202212131615 Solution to 1989-CE-AMATH-I-6

$p$ and $q$ are real numbers such that

$(p+qi)^2=21-20i$ .

Find the values of $p$ and $q$ . Hence write down the two square roots of $21-20i$ .

Roughwork.

$\begin{aligned} \textrm{LHS} & = (p+qi)^2 \\ & = (p^2-q^2)+(2pq)i \\ \textrm{RHS} & = (21)+(-20)i \\ \end{aligned}$

$\begin{cases} p^2-q^2=21 \\ pq = -10 \end{cases}$

$\therefore$ $(p,q)=(\pm 5,\mp 2)$ .

物理後生﹑掣數學之肘.
小子算數﹑指物理之月.