Square Root of Complex Number in Cartesian Form/Examples/3+4i

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Example of Square Root of Complex Number in Cartesian Form

$\sqrt {3 + 4 i} = \pm \left({2 + i}\right)$


Proof

\(\ds \left({x + i y}\right)^2\) \(=\) \(\ds 3 + 4 i\)
\(\ds \leadsto \ \ \) \(\ds x^2\) \(=\) \(\ds \dfrac {3 + \sqrt {3^2 + 4^2} } 2\) Square Root of Complex Number in Cartesian Form
\(\ds \) \(=\) \(\ds \dfrac {3 + \sqrt {25} } 2\)
\(\ds \) \(=\) \(\ds \dfrac {3 + 5} 2\)
\(\ds \) \(=\) \(\ds 4\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \pm 2\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds \pm \dfrac 4 {2 \times 2}\)
\(\ds \) \(=\) \(\ds \pm 1\)


As $2 x y = 4$ it follows that the two solutions are:

$2 + i$
$-2 - i$


Sources