1. **State the problem:** 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 can use factoring, completing the square, or the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Important: The discriminant $$\Delta = b^2 - 4ac$$ determines the nature of the roots. 3. **Apply factoring method:** Look for two numbers that multiply to $$6$$ (the constant term) and add to $$-5$$ (the coefficient of $$x$$). These numbers are $$-2$$ and $$-3$$ because $$-2 \times -3 = 6$$ and $$-2 + (-3) = -5$$. 4. **Factor the quadratic:** $$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$$ 5. **Solve for $$x$$:** Set each factor equal to zero: $$x - 2 = 0 \Rightarrow x = 2$$ $$x - 3 = 0 \Rightarrow x = 3$$ 6. **Final answer:** The solutions to the equation are $$x = 2$$ and $$x = 3$$.