1. Problem: Solve the quadratic equation $x^2 - 5x + 6 = 0$ and find its roots. 2. Formula: The roots of a quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Important rules: - The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots. - If $\Delta > 0$, two distinct real roots. - If $\Delta = 0$, one real root (repeated). - If $\Delta < 0$, no real roots (complex roots). 3. Identify coefficients: $a=1$, $b=-5$, $c=6$. 4. Calculate discriminant: $$\Delta = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ 5. Since $\Delta = 1 > 0$, there are two distinct real roots. 6. Calculate roots: $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 7. Roots: - $x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3$ - $x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2$ 8. Final answer: The roots of the equation $x^2 - 5x + 6 = 0$ are $x=3$ and $x=2$.