Question

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

A
±(2+5i)
B
±(25i)
C
±(4+3i)
D
±(43i)

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 $$\pm(2-5i)$$Maths

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