1. **State the problem:** Solve the quadratic equation $$x^2 - 10x = -9$$ by completing the square. 2. **Formula and rule:** To complete the square for an equation of the form $$x^2 + bx = c$$, add $$\left(\frac{b}{2}\right)^2$$ to both sides to form a perfect square trinomial. 3. **Identify terms:** Here, $$b = -10$$. 4. **Calculate the term to add:** $$\left(\frac{-10}{2}\right)^2 = (-5)^2 = 25$$. 5. **Add 25 to both sides:** $$x^2 - 10x + 25 = -9 + 25$$ 6. **Simplify the right side:** $$x^2 - 10x + 25 = 16$$ 7. **Rewrite the left side as a square:** $$ (x - 5)^2 = 16 $$ 8. **Take the square root of both sides:** $$ \sqrt{(x - 5)^2} = \pm \sqrt{16} $$ $$ |x - 5| = \pm 4 $$ 9. **Solve for $$x$$:** $$ x - 5 = 4 \quad \text{or} \quad x - 5 = -4 $$ $$ x = 9 \quad \text{or} \quad x = 1 $$ **Final answer:** $$x = 9$$ or $$x = 1$$