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Question

Find the square root of complex number $$- 21 - 20i$$.


A
±(2+5i)
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B
±(25i)
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C
±(4+3i)
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D
±(43i)
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Solution

The correct option is B $$\pm(2-5i)$$

Let $$\sqrt{-21-20i}=x+iy$$

$$\implies -21-20i=(x+iy)^2=x^2-y^2+2ixy$$

Equating real and imaginary parts we get

$$x^2-y^2=-21$$ ------(1)

$$2xy=-20$$ ------(2)

Therefore, $$(x^2+y^2)^2=(x^2-y^2)^2+4x^2y^2=(-21)^2+(-20)^2=841$$

$$\implies x^2+y^2=\sqrt{841}=29$$ ------(3)

adding (1) and (3) we get

$$2x^2=8$$

$$\implies x=\pm\sqrt{4}=\pm 2$$

Substituting $$x$$ in (2) we get

$$y=\mp 5$$

Therefore, the square root of $$-21-20i$$ is $$\pm(2-5i)$$

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