Find the square root of the following complex numbers: 7 + 24i - Vidyadip

Question

Find the square root of the following complex numbers:
7 + 24i

✓

Answer

Let $\sqrt{7+24 i}=\mathrm{a}+\mathrm{bi}$, where $\mathrm{a}, \mathrm{b} \in \mathrm{R}$ Squaring on both sides, we get
$
\begin{aligned}
& 7+24 i=(a+b i)^2 \\
& 7+24 i=a^2+b^2 i^2+2 a b i \\
& 7+24 i=\left(a^2-b^2\right)+2 a b i \quad \ldots . .\left[\because i^2=-1\right]
\end{aligned}
$
Equating real and imaginary parts, we get
$
\begin{array}{ll}
& a^2-b^2=7 \text { and } 2 a b=24 \\
\therefore & a^2-b^2=7 \text { and } b=\frac{12}{a} \\
\therefore & a^2-\left(\frac{12}{a}\right)^2=7 \\
\therefore & a^2-\frac{144}{a^2}=7 \\
\therefore & a^4-144=7 a^2 \\
\therefore & a^4-7 a^2-144=0 \\
\therefore & \left(a^2-16\right)\left(a^2+9\right)=0 \\
\therefore & a^2=16 \text { or } a^2=-9 \\
\therefore & \text { But } a \in R \\
\therefore & a^2 \neq-9 \\
\therefore & a^2=16 \\
& a= \pm 4 \text { } \\
& \text { When } a=4, b=\frac{12}{4}=3 \\
& \text { When } a=-4, b=\frac{12}{-4}=-3 \\
\therefore & \sqrt{7+24 i}= \pm(4+3 i)
\end{array}
$

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