Finding solutions to a complex polynomial

Question

I have the quadratic equation $x^2-5x+7-i=0$, I am not sure how to solve? I was going to use the quadratic formula but I wasn't sure as the quadratic equation seems quite easy and working out $\sqrt{-3+4i}$ seemed liked I was going too far?

Would using sum of roots and product of roots be better?

Thank you!

Answer 1

Hint: notice that $$(2i+1)^2 = -3+4i$$

Answer 2

You are on the right track. Now, use the fact that $(1+2i)^2=-3+4i$.