1. **State the problem:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Recall the formula:** For a quadratic equation $$ax^2 + bx + c = 0$$, the solutions can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-5$, and $c=6$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ Since $\Delta > 0$, there are two distinct real roots. 4. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 5. **Find the two solutions:** - For the plus sign: $$x = \frac{5 + 1}{2} = \frac{6}{2} = 3$$ - For the minus sign: $$x = \frac{5 - 1}{2} = \frac{4}{2} = 2$$ 6. **Final answer:** The solutions to the equation are $$x = 3$$ and $$x = 2$$.