1. **State the problem:** Solve the quadratic equation $x^2 - 3x = -2$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 3x + 2 = 0$$ 3. **Identify the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Apply the formula:** Here, $a=1$, $b=-3$, and $c=2$. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-3)^2 - 4 \times 1 \times 2 = 9 - 8 = 1$$ 5. **Calculate the roots:** $$x = \frac{-(-3) \pm \sqrt{1}}{2 \times 1} = \frac{3 \pm 1}{2}$$ 6. **Find the two solutions:** $$x_1 = \frac{3 + 1}{2} = \frac{4}{2} = 2$$ $$x_2 = \frac{3 - 1}{2} = \frac{2}{2} = 1$$ 7. **Final answer:** The solutions to the equation are $x=2$ and $x=1$.