find square roots of -1+2i

Tuesday, November 29, 2016

We have to find the square root of $\displaystyle-{1}+{2}{i}$ i.e. $\sqrt{-1+2i}$

We will find the square roots of the complex number of the form x+yi , where x and y are real numbers, by the following method:

Let $\displaystyle{{z}}^{{2}}={{\left({x}+{y}{i}\right)}}^{{2}}=-{1}+{2}{i}$

i.e. $\displaystyle{\left({{x}}^{{2}}-{{y}}^{{2}}\right)}+{2}{x}{y}{i}=-{1}+{2}{i}$

Comparing real and imaginary terms we get,

$\displaystyle{{x}}^{{2}}-{{y}}^{{2}}=-{1}------\to{\left({1}\right)}$

$\displaystyle{2}{x}{y}={2}$ implies $\displaystyle{x}{y}={1}-----\to{\left({2}\right)}$

So from (2) we get, y=1/x . Substituting this in (1) we have,

$x^2-\frac{1}{x^2}=-1$

i.e. $\displaystyle{{x}}^{{4}}+{{x}}^{{2}}-{1}={0}$

implies $x^2=\frac{-1\pm\sqrt{5}}{2}$

$\displaystyle={0.62},-{1.62}$

Therefore, $x=\pm\sqrt{0.62}=\pm 0.79$

$\displaystyle{{x}}^{{2}}=-{1.62}$ is discarded since it gives imaginary value.

hence,

When x=0.79, y= 1.27

x=-0.79 , y= -1.27

i.e we have, $\sqrt{-1+2i}=0.79+1.27i or -0.79-1.27i$

$=\pm (0.79+1.27i)$

Hence the square roots of -1+2i are: $\pm$ (0.79+1.27i)

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