What is the square root of z = 7 - 24i?

Correct option is D

​$$\begin{aligned}&\text{Let } z = 7 - 24i. \text{ We are looking for } \sqrt{z} = a + bi, \text{ such that:} \\&(a + bi)^2 = 7 - 24i \\[5pt]&\text{Expanding the left-hand side:} \\&(a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi \\[5pt]&\text{Equating real and imaginary parts:} \\&a^2 - b^2 = 7 \quad \text{(1)} \\&2ab = -24 \quad \text{(2)} \\[5pt]&\text{From equation (2):} \\&ab = -12 \Rightarrow b = \frac{-12}{a} \\[5pt]&\text{Substitute into equation (1):} \\&a^2 - \left( \frac{-12}{a} \right)^2 = 7 \Rightarrow a^2 - \frac{144}{a^2} = 7 \\[5pt]&\text{Multiply both sides by } a^2: \\&a^4 - 144 = 7a^2 \Rightarrow a^4 - 7a^2 - 144 = 0 \\[5pt]&\text{Let } u = a^2, \text{ then:} \\&u^2 - 7u - 144 = 0 \\[5pt]&\text{Solve using the quadratic formula:} \\&u = \frac{7 \pm \sqrt{49 + 576}}{2} = \frac{7 \pm \sqrt{625}}{2} = \frac{7 \pm 25}{2} \\[5pt]&u = 16 \text{ or } -9 \Rightarrow a^2 = 16 \Rightarrow a = \pm 4 \\[5pt]&\text{Using } ab = -12: \\&\text{If } a = 4, \text{ then } b = \frac{-12}{4} = -3 \\&\text{If } a = -4, \text{ then } b = \frac{-12}{-4} = 3 \\[5pt]&\text{Thus, the square roots of } 7 - 24i \text{ are:} \\&\boxed{\pm (4 - 3i)}\end{aligned}$$​​