1. **Problem Statement:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula and Rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the equation. 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. **Evaluate the roots:** $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 6. **Find the two solutions:** - For the plus sign: $$x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3$$ - For the minus sign: $$x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2$$ 7. **Answer:** The solutions to the equation are $$x = 3$$ and $$x = 2$$.