What should be my thought process in simplifying...?

Question

(x^2-2x+2)=0
The answer's (1+i)(1-i), but I'm not sure how I'd do it in my head. Any tips on simplifying polynomials with imaginary roots?

Answer 1

x^2 - 2x + 2 = 0

The direct way would be to use the quadratic formula. Another way would be to complete the square, like so:

x^2 - 2x + 1 + 1 = 0

x^2 - 2x + 1 = -1

(x - 1)^2 = -1

Take the square root of both sides (remembering to add a "plus or minus"):

x - 1 = +/- sqrt(-1)

But sqrt(-1) = i, so

x - 1 = +/- i

Moving the -1 to the right hand side,

x = 1 +/- i

therefore,

x = (1 + i, 1 - i)

Answer 2

You could use the discriminant first to determine that the roots are imaginary and then the quadratic formula would be the easiest way to go.