1. The problem is to solve the quadratic equation $x^2 - 5x + 6 = 0$. 2. The formula to solve quadratic equations is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. For the equation $x^2 - 5x + 6 = 0$, the coefficients are $a=1$, $b=-5$, and $c=6$. 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ 5. Since the discriminant is positive, there are two real solutions. 6. Substitute values into the quadratic formula: $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 7. Calculate the two solutions: $$x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3$$ $$x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2$$ 8. Therefore, the solutions to the equation are $x=3$ and $x=2$. This means the quadratic equation factors as $(x-3)(x-2)=0$. Final answer: $x=3$ or $x=2$.