Chapter 9.8, Problem 22WE

To determine

To calculate: To find the square roots of complex number

Answer to Problem 22WE

Square roots of complex number are $\left(4+3i\right)$ and $\left(-4-3i\right)$

Explanation of Solution

Given information: Complex number is $7-24i$

Calculation:

Given the complex number as follows:

  $7-24i$

Let the square root of complex number be $x+yi$

Thus,

  $\begin{array}{l}{\left(x+yi\right)}^2=7-24i\\ {\left(x\right)}^2+{\left(yi\right)}^2-2xyi=7-24i\\ \left(x^2-y^2\right)-2xyi=7-24i\end{array}$

Equating the real and imaginary components:

  $\begin{array}{l}{x}^2-y^2=7---(1)\\ -2xy=24----(2)\end{array}$

From equation (2),

  $\begin{array}{l}-2xy=-24\\ x=\frac{12}{y}\end{array}$

Substituting the value in (1),

  $\begin{array}{l}{\left(\frac{12}{y}\right)}^2-y^2=7\\ \frac{1444}{y^2}-y^2=7\\ 1444-y^4=7y^2\\ y^4+7y^2-144=0\\ y^4+16y^2-9y^2-4=0\\ \left(y^2+16\right)\left(y^2-9\right)=0\\ y^2=-16,9\end{array}$

  $y^2\ne -16$ as its imaginary

Thus, $y^2=9$

  $y=\pm 3$

When $y=3$

  $\begin{array}{l}x=\frac{12}{3}\\ x=4\end{array}$

When $y=-3$

  $\begin{array}{l}x=\frac{12}{\left(-3\right)}\\ x=-4\end{array}$

Conclusion:

Hence, square roots of complex number are $\left(4+3i\right)$ and $\left(-4-3i\right)$