1. **State the problem:** Solve the equation $x^2 - 5x + 6 = 0$. 2. **Formula used:** For quadratic equations of the form $ax^2 + bx + c = 0$, the solutions are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 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 $+$ sign: $x = \frac{5 + 1}{2} = \frac{6}{2} = 3$ - For $-$ sign: $x = \frac{5 - 1}{2} = \frac{4}{2} = 2$ 7. **Final answer:** The solutions to the equation are $x=3$ and $x=2$. This means the quadratic factors as $(x-3)(x-2)=0$. These steps show how to solve any quadratic equation using the quadratic formula, including calculating the discriminant to determine the nature of the roots.