How to find the complex roots of $x^2-2ax+a^2+b^2$?

Question

How to find the complex roots of $x^2-2ax+a^2+b^2$?

I tried using the quadratic formula:

$$ x_{1,2} = \frac{2a \pm \sqrt {4a^2-4b^2}}{2} = {a \pm \sqrt {a^2-b^2}} = a\pm \sqrt{a-b}\sqrt{a+b}$$

I tried to represent each root as $x+iy$ but got stuck along the way. What should I do?

Thanks

Answer 1

Your equation can be rewritten as $$(x-a)^2=-b^2\iff x-a=\pm ib$$ so the roots are $x_{1,2}=a\pm ib$.

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The quadratic formula also gives the right answer, but it should be $$x_{1,2}=\frac{2a\pm\sqrt{4a^2-4(a^2+b^2)}}2=a\pm\sqrt{-b^2}=a\pm ib$$

Answer 2

$x^2 - 2ax + a^2 + b^2 = 0 \Rightarrow (x - a)^2 = -b^2 \Rightarrow x = a \pm ib.$