1. **State the problem:** Solve the quadratic equation $$x^2 - 9x + 20 = 0$$. 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. Here, factoring is suitable because the coefficients are integers. 3. **Factoring the quadratic:** We look for two numbers that multiply to $$20$$ (the constant term) and add to $$-9$$ (the coefficient of $$x$$). 4. The numbers $$-4$$ and $$-5$$ satisfy this because $$-4 \times -5 = 20$$ and $$-4 + -5 = -9$$. 5. Rewrite the quadratic as: $$x^2 - 4x - 5x + 20 = 0$$ 6. Group terms: $$(x^2 - 4x) + (-5x + 20) = 0$$ 7. Factor each group: $$x(x - 4) - 5(x - 4) = 0$$ 8. Factor out the common binomial: $$(x - 5)(x - 4) = 0$$ 9. Set each factor equal to zero: $$x - 5 = 0 \quad \Rightarrow \quad x = 5$$ $$x - 4 = 0 \quad \Rightarrow \quad x = 4$$ 10. **Final answer:** The solutions to the equation are $$x = 4$$ and $$x = 5$$.