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

Example of Square Root of Complex Number in Cartesian Form

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

$\ds \paren {x + i y}^2$

$=$

$\ds 5 - 12 i$

$\ds \leadsto \ \ $

$\ds x^2$

$=$

$\ds \dfrac {5 + \sqrt {5^2 + \paren {-12}^2} } 2$

$\ds $

$=$

$\ds \dfrac {5 + \sqrt {169} } 2$

$\ds $

$=$

$\ds \dfrac {5 + 13} 2$

$\ds $

$=$

$\ds 9$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds \pm 3$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds \pm \dfrac {-12} {2 \times 3}$

$\ds $

$=$

$\ds \mp 2$

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

$3 - 2 i$

$-3 + 2 i$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Example
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $98 \ \text{(a)}$