Question

Find the square roots of $5+12i$

• A. $\sqrt{5+12i}=\pm (3+2i)$.
• B. $\sqrt{5+12i}=\pm (3-2i)$.
• C. $\sqrt{5+12i}=\pm (2+3i)$.
• D. $\sqrt{5+12i}=\pm (2-3i)$.

Answer 1

The correct option is A $\sqrt{5+12i}=\pm (3+2i)$.

Let $\sqrt{5+12i}=x+iy$.
Then, $5+12i=(x+iy{)}^2$
or $5+12i=(x^2-y^2)+2ixy$
or $x^2-y^2=5$ (1)
and
$2xy=12$ (2)
Now,
$(x^2+y^2{)}^2=(x^2-y^2{)}^2+4x^2{y}^2$
or $(x^2+y^2{)}^2=5^2+{12}^2=169$
or $x^2+y^2=13(\therefore x^2+y^2>0)$ (3)
On solving (1) and (3), we get
$x^2=9$ and $y^2=4⟹x=\pm 3$ and $y=\pm 2$
From (2), 2xy is positive. So, x and y are of the same sign. Hence,
$x=3$ and $y=2$ or $x=-3$ and $y=-2$
Hence, $\sqrt{5+12i}=\pm (3+2i)$.
Ans: A