Question

If the square root of $-7+24i$is $x+iy$ then $x$ is equal to

• A. $\pm 1$
• B. $\pm 2$
• C. $\pm 3$
• D. $\pm 4$

Answer 1

The correct option is C

$\pm 3$

Explanation for the correct option

Given, square root of $-7+24i$is $x+iy$.

Let $\sqrt{-7+24i}=x+iy$
On squaring, we get
$-7+24i=\left(x^2-y^2\right)+2ixy$
Equating the real and imaginary parts
$\begin{array}{rcl}{x}^2-y^2& =& -7\text{and}2xy=24\\ x^2+y^2& =& \sqrt{\left(x^2-y^2\right)+4x^2{y}^2}\\ & =& \sqrt{(-7{)}^2+(24{)}^2}\\ & =& 25\end{array}$
Solving, $x^2-y^2=-7$ and $x^2+y^2=25$ we get :

$x^2=9$ and $y^2=16$
$\therefore \mathrm{x}=\pm 3$ and $\mathrm{y}=\pm 4$
Hence, option C is correct.