1. **State the problem:** Solve the quadratic equation $$x^2 - 6x - 16 = 0$$ using the completing the square method. 2. **Recall the formula and rule:** To complete the square for an equation of the form $$x^2 + bx + c = 0$$, we rewrite it as $$\left(x - \frac{b}{2}\right)^2 = \text{some number}$$ by adding and subtracting $$\left(\frac{b}{2}\right)^2$$. 3. **Rewrite the equation:** Move the constant term to the right side: $$x^2 - 6x = 16$$ 4. **Complete the square:** Take half of the coefficient of $$x$$, which is $$-6$$, so half is $$-3$$, and square it: $$\left(-3\right)^2 = 9$$ Add and subtract 9 on the left side: $$x^2 - 6x + 9 - 9 = 16$$ Rewrite grouping the perfect square trinomial: $$\left(x - 3\right)^2 - 9 = 16$$ 5. **Isolate the perfect square:** $$\left(x - 3\right)^2 = 16 + 9$$ $$\left(x - 3\right)^2 = 25$$ 6. **Take the square root of both sides:** $$x - 3 = \pm \sqrt{25}$$ $$x - 3 = \pm 5$$ 7. **Solve for $$x$$:** $$x = 3 \pm 5$$ This gives two solutions: $$x = 3 + 5 = 8$$ $$x = 3 - 5 = -2$$ **Final answer:** $$x = 8, -2$$