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  4. If (-7-24 i)1 / 2=x-i y, then x2+y2=

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Q. If $(-7-24 i)^{1 / 2}=x-i y$, then $x^2+y^2=$

Complex Numbers and Quadratic Equations

Solution:

$z=(-7-24 i)^{1 / 2}$
$z^2=-7-24 i$
$z=x-i y$
$z^2=x^2-y^2-2 i x y$
$x^2-y^2=-7$
$2 x y=24$
$x y=12$
$\left(x^2-y^2\right)^2+4 x^2 y^2=\left(x^2+y^2\right)^2$
$49+4.144=\left(x^2+y^2\right)^2$
$\left(x^2+y^2\right)^2=25^2$
$x^2+y^2=25$