I have to solve each equation and verify my solution graphically... but first, I would like to create the piecewise for $|x^2 - 2x + 2| = 3x - 4$ to have a reference to know if my solution is extraneous.
I know that I would need to find the solutions for $x^2 - 2x + 2$ (Case 1) and $-x^2 + 2x - 2$ (Case 2), and since it's not factorable, I used the quadratic formula:
[Case 1] \begin{align*} x & = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(2)}}{2}\\ x & = \frac{2 \pm \sqrt{4 - 8}}{2} \end{align*} This is where I got stuck... I'd get $-12$ in the square root which isn't correct, but my math and variables were right. I don't understand what I did wrong?
[edit] Same thing happened when solving Case 2, where I got $-4$ in the square root.
Note that $x^2-2x+2=(x-1)^2+1$ is always positive, so you can just remove the absolute value bars. Then $$x^2-2x+2=3x-4\\x^2-5x+6=0\\(x-2)(x-3)=0\\x=2,3$$