1. **State the problem:** We are given a quadratic equation $$x^2 + 12x + k = 0$$ and told that one root is twice the other. We need to find the value of $$k$$. 2. **Recall the relationships between roots and coefficients:** For a quadratic equation $$ax^2 + bx + c = 0$$ with roots $$r_1$$ and $$r_2$$, we have: - Sum of roots: $$r_1 + r_2 = -\frac{b}{a}$$ - Product of roots: $$r_1 r_2 = \frac{c}{a}$$ 3. **Apply to our equation:** Here, $$a=1$$, $$b=12$$, and $$c=k$$. So: - $$r_1 + r_2 = -12$$ - $$r_1 r_2 = k$$ 4. **Use the condition that one root is twice the other:** Let the smaller root be $$r$$, then the other root is $$2r$$. 5. **Write equations using this:** - Sum: $$r + 2r = 3r = -12 \implies r = -4$$ - Product: $$r \times 2r = 2r^2 = k$$ 6. **Calculate $$k$$:** - $$k = 2 \times (-4)^2 = 2 \times 16 = 32$$ **Final answer:** $$k = 32$$