Q.Find the square root of the complex number $7 + 24i$.

Write $\sqrt{7+24i}=x+iy$, square both sides, match real and imaginary parts, and solve for $x,y$ using the modulus as a third equation.

Let $\sqrt{7+24i} = x+iy$ where $x,y$ are real. Squaring,

$x^2 - y^2 + 2ixy = 7+24i.$

Comparing real and imaginary parts:

$x^2-y^2 = 7 \qquad \text{and} \qquad 2xy = 24 \Rightarrow xy = 12.$

A third relation comes from equating moduli: $|x+iy|^2 = |7+24i|$, i.e.

$x^2+y^2 = \sqrt{7^2+24^2} = \sqrt{49+576} = \sqrt{625} = 25.$

…