1. Let's state the problem: find the value(s) of $k$ such that the quadratic equation $$x^2 + 3x - k = 0$$ has one repeated root. 2. A quadratic equation has one repeated root when its discriminant is zero. The discriminant $\Delta$ is given by: $$\Delta = b^2 - 4ac$$ where $a$, $b$, and $c$ are the coefficients of the equation $ax^2 + bx + c = 0$. 3. In our equation, $a = 1$, $b = 3$, and $c = -k$. 4. Calculate the discriminant: $$\Delta = 3^2 - 4(1)(-k) = 9 + 4k$$ 5. Set the discriminant equal to zero because the root is repeated: $$9 + 4k = 0$$ 6. Solve for $k$: $$4k = -9$$ $$k = -\frac{9}{4}$$ 7. Therefore, the quadratic equation has one repeated root when $$k = -\frac{9}{4}$$.