1. **State the problem:** Solve the equation $ (x-2)^2 - 9 = 0 $ for $ x $. 2. **Use the formula:** Recognize this as a quadratic equation in the form $ (a)^2 - b^2 = 0 $, which can be factored using the difference of squares formula: $$ a^2 - b^2 = (a - b)(a + b) $$ Here, $ a = (x-2) $ and $ b = 3 $ because $ 9 = 3^2 $. 3. **Apply the difference of squares:** $$ (x-2)^2 - 3^2 = ((x-2) - 3)((x-2) + 3) = 0 $$ 4. **Set each factor equal to zero:** $$ (x-2) - 3 = 0 \quad \text{or} \quad (x-2) + 3 = 0 $$ 5. **Solve each equation:** - For $ (x-2) - 3 = 0 $: $$ x - 2 - 3 = 0 $$ $$ x - \cancel{2} - 3 + \cancel{2} = 0 + 2 $$ $$ x - 3 = 2 $$ $$ x = 5 $$ - For $ (x-2) + 3 = 0 $: $$ x - 2 + 3 = 0 $$ $$ x + 1 = 0 $$ $$ x = -1 $$ 6. **Final answer:** The solutions to the equation are $$ x = 5 \quad \text{or} \quad x = -1 $$