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

## 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:

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

$\blacksquare$