Complex Quadratic

Problem:



Solution:

We will use the "completing the square" method to solve this problem.

First. note that $ (x = 1)^2 = x^2  - 2x + 1 $ (by simply multiplying).
We can get

$ x^2 - 2x + 2 =  x^2 - 2x + 1 + 1 = (x - 1)^2 + 1 = 0 $

Rearranging, we get

$ (x - 1)^2 = -1 $.

This implies the solution has no real solution because no real number's square is a negative number. But if you knew complex number, you will recognize $ (x - 1) = \pm i $, where $ i $ is the imaginary unit, so the two complex solutions are $ 1 \pm i $.