Question

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

• A. $\pm (2+5i)$
  
• B. $\pm (2-5i)$
  
• C. $\pm (4+3i)$
  
• D. $\pm (4-3i)$
  

Answer 1

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

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

$⟹-21-20i=(x+iy{)}^2=x^2-y^2+2ixy$

Equating real and imaginary parts we get

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

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

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

$⟹x^2+y^2=\sqrt{841}=29$ ------(3)

adding (1) and (3) we get

$2x^2=8$

$⟹x=\pm \sqrt{4}=\pm 2$

Substituting $x$ in (2) we get

$y=\mp 5$

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