1. **Problem statement:** Solve the equation $$(x - 2)(x - 3) = 20$$ for $x$. 2. **Formula and rules:** To solve this, first expand the left side using the distributive property (FOIL method): $$ (x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6 $$ 3. **Rewrite the equation:** $$ x^2 - 5x + 6 = 20 $$ 4. **Bring all terms to one side to set the equation to zero:** $$ x^2 - 5x + 6 - 20 = 0 $$ $$ x^2 - 5x - 14 = 0 $$ 5. **Solve the quadratic equation:** Use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=-5$, and $c=-14$. 6. **Calculate the discriminant:** $$ \Delta = b^2 - 4ac = (-5)^2 - 4(1)(-14) = 25 + 56 = 81 $$ 7. **Find the roots:** $$ x = \frac{-(-5) \pm \sqrt{81}}{2(1)} = \frac{5 \pm 9}{2} $$ 8. **Evaluate each root:** - For $+$: $$ x = \frac{5 + 9}{2} = \frac{14}{2} = 7 $$ - For $-$: $$ x = \frac{5 - 9}{2} = \frac{-4}{2} = -2 $$ 9. **Final answer:** $$ x = 7 \quad \text{or} \quad x = -2 $$