1. **Stating the problem:** Solve the quadratic equation $x^2 - 5x + 6 = 0$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 1$, $b = -5$, and $c = 6$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 6. **Find the two solutions:** - For the plus sign: $$x = \frac{5 + 1}{2} = \frac{6}{2} = 3$$ - For the minus sign: $$x = \frac{5 - 1}{2} = \frac{4}{2} = 2$$ 7. **Final answer:** The solutions to the quadratic equation are $x = 3$ and $x = 2$.