Find square root of -15-8i ?

\pm (1 - 4 i)

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\pm (1 + 4 i)

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\pm (- 1 - 4 i)

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\pm (- 1 + 4 i)

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Solution

The correct options are
A $\pm (1 - 4 i)$
C $\pm (- 1 + 4 i)$
Let the square root of $- 15 - 8 i$ be $x + i y .$
$\Rightarrow x^{2} - y^{2} + 2 i x y = - 15 - 8 i$
$\Rightarrow x y = - 4$ and $x^{2} - y^{2} = - 15$
So, $x^{2} - \frac{16}{x^{2}} = - 15$
$x^{4} - 16 + 15 x^{2} = 0$
$\Rightarrow (x^{2} - 1) (x^{2} + 16) = 0$
So, $x = \pm 1$ . Similarly, $y = \pm 4$ correspondingly using $x y = - 4.$

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