1. **State the problem:** Solve the quadratic equation $$w^2 - 2w - 13 = 0$$ by completing the square. 2. **Recall the formula and method:** To complete the square for an equation of the form $$w^2 + bw + c = 0$$, we rewrite it as $$\left(w + \frac{b}{2}\right)^2 = \text{some number}$$ by adding and subtracting the square of half the coefficient of $w$. 3. **Rewrite the equation:** $$w^2 - 2w - 13 = 0$$ Move the constant term to the right side: $$w^2 - 2w = 13$$ 4. **Complete the square:** Take half of the coefficient of $w$, which is $-2$, half is $-1$, and square it: $$\left(-1\right)^2 = 1$$ Add and subtract 1 on the left side: $$w^2 - 2w + 1 - 1 = 13$$ Group the perfect square trinomial: $$\left(w - 1\right)^2 - 1 = 13$$ 5. **Isolate the perfect square:** $$\left(w - 1\right)^2 = 13 + 1$$ $$\left(w - 1\right)^2 = 14$$ 6. **Take the square root of both sides:** $$w - 1 = \pm \sqrt{14}$$ 7. **Solve for $w$:** $$w = 1 \pm \sqrt{14}$$ 8. **Calculate approximate decimal values:** $$\sqrt{14} \approx 3.74$$ So, $$w \approx 1 + 3.74 = 4.74$$ $$w \approx 1 - 3.74 = -2.74$$ **Final answer:** $$w \approx 4.74 \text{ or } w \approx -2.74$$