Solve the problem using completing the squares (1)X² − 3X+3=0 and (2)−X² +2X−2=0.

Take half of the coefficient of $X$ (which is $-3$), square it $\left(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\right)$, and add it to both sides of the equation.

$$X^2 - 3X + \frac{9}{4} = -3 + \frac{9}{4}$$

$$\left(X - \frac{3}{2}\right)^2 = -\frac{12}{4} + \frac{9}{4}$$ $$\left(X - \frac{3}{2}\right)^2 = -\frac{3}{4}$$

$$X - \frac{3}{2} = \pm\sqrt{-\frac{3}{4}}$$ $$X - \frac{3}{2} = \pm\frac{\sqrt{3}}{2}i$$

$$X = \frac{3}{2} \pm \frac{\sqrt{3}}{2}i$$ $$X = \frac{3 \pm i\sqrt{3}}{2}$$

The solutions are $\boxed{X = \frac{3 + i\sqrt{3}}{2}, X = \frac{3 - i\sqrt{3}}{2}}$.

2) Solve $-X^2 + 2X - 2 = 0$

Take half of the coefficient of $X$ (which is $-2$), square it $\left((-1)^2 = 1\right)$, and add it to both sides of the equation.

$$X^2 - 2X + 1 = -2 + 1$$

$$X - 1 = \pm\sqrt{-1}$$ $$X - 1 = \pm i$$

$$X = 1 \pm i$$

The solutions are $\boxed{X = 1 + i, X = 1 - i}$.