Square Root of Complex Number in Cartesian Form/Examples/5-12i

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Example of Square Root of Complex Number in Cartesian Form

$\sqrt {5 - 12 i} = \pm \paren {3 - 2 i}$


Proof

\(\displaystyle \paren {x + i y}^2\) \(=\) \(\displaystyle 5 - 12 i\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2\) \(=\) \(\displaystyle \dfrac {5 + \sqrt {5^2 + \paren {-12}^2} } 2\) Square Root of Complex Number in Cartesian Form
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {5 + \sqrt {169} } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {5 + 13} 2\)
\(\displaystyle \) \(=\) \(\displaystyle 9\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle \pm 3\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle \pm \dfrac {-12} {2 \times 3}\)
\(\displaystyle \) \(=\) \(\displaystyle \mp 2\)

As $2 x y = -12$ it follows that the two solutions are:

$3 - 2 i$
$-3 + 2 i$

$\blacksquare$


Sources