The square root of 3 -4i is

MCQ

The square root of $3 -4i$ is

✓
$ \pm (2 + i)$
B
$ \pm (2 - i)$
C
$ \pm (1 - 2i)$
D
$ \pm (1 + 2i)$

✓

Answer

Correct option: A.

$ \pm (2 + i)$

a
(a) Let $\sqrt {3 - 4i} = x + iy$$ \Rightarrow \,\,3 - 4i = \,{x^2} - {y^2} + 2ixy$
$ \Rightarrow {x^2} - {y^2} = 3,$ $2xy = - 4$ ......$(i)$
$ \Rightarrow \,\,{({x^2} + {y^2})^2} = \,{({x^2} - {y^2})^2} + 4{x^2}{y^2}$$ = {(3)^2} + {( - 4)^2} = 25$
$ \Rightarrow \,{x^2} + {y^2} = 5$…..$(ii)$
From equation $(i) $ and $(ii) $ ${x^2} = 4\, \Rightarrow \,x = \pm \,2$,
${y^2} = 1$$ \Rightarrow \,y = \, \pm \,1.$Hence the square root of $(3 - 4i)$ is $\, \pm \,(2 + i)$.

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