Square Root of Complex Number in Cartesian Form/Examples/-8+6i

Example of Square Root of Complex Number in Cartesian Form

$\sqrt {-8 + 6 i} = \pm \paren {1 + 3 i}$

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

$=$

$\ds -8 + 6 i$

$\ds \leadsto \ \ $

$\ds x^2$

$=$

$\ds \dfrac {-8 + \sqrt {\paren {-8}^2 + 6^2} } 2$

$\ds $

$=$

$\ds \dfrac {-8 + \sqrt {100} } 2$

$\ds $

$=$

$\ds \dfrac {-8 + 10} 2$

$\ds $

$=$

$\ds 1$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds \pm 1$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds \pm \dfrac 6 {2 \times 1}$

$\ds $

$=$

$\ds \pm 3$

As $2 x y = 6$ it follows that the two solutions are:

$1 + 3 i$

$-1 - 3 i$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm: Examples: $\text {(iii)}$