1. **State the problem:** We are given the quadratic equation $$(X - k)^2 - 6X = 0$$ and told that one root is the additive inverse of the other. We need to find the value of $k$. 2. **Rewrite the equation:** Expand $$(X - k)^2$$: $$ (X - k)^2 = X^2 - 2kX + k^2 $$ So the equation becomes: $$ X^2 - 2kX + k^2 - 6X = 0 $$ 3. **Combine like terms:** $$ X^2 - (2k + 6)X + k^2 = 0 $$ 4. **Recall the properties of roots:** For a quadratic equation $$ax^2 + bx + c = 0$$ with roots $$r_1$$ and $$r_2$$, - Sum of roots $$r_1 + r_2 = -\frac{b}{a}$$ - Product of roots $$r_1 r_2 = \frac{c}{a}$$ 5. **Apply the condition:** One root is the additive inverse of the other, so if one root is $$r$$, the other is $$-r$$. - Sum of roots: $$r + (-r) = 0$$ - Product of roots: $$r \times (-r) = -r^2$$ 6. **Use sum of roots from the equation:** $$ r_1 + r_2 = -\frac{b}{a} = -\frac{-(2k + 6)}{1} = 2k + 6 $$ Since sum of roots is 0, $$ 2k + 6 = 0 $$ 7. **Solve for $k$:** $$ 2k = -6 $$ $$ k = -3 $$ **Final answer:** $$ \boxed{-3} $$