1. The problem asks us to find the value(s) of $k$ such that the quadratic equation $$x^2 - 3x + k = 0$$ has a repeated root. 2. A quadratic equation has a repeated root when its discriminant is zero. The discriminant $\Delta$ is given by $$\Delta = b^2 - 4ac,$$ where $a$, $b$, and $c$ are coefficients from the equation $ax^2 + bx + c = 0$. 3. In our equation, $a = 1$, $b = -3$, and $c = k$. Substitute these into the discriminant formula: $$\Delta = (-3)^2 - 4 \times 1 \times k = 9 - 4k.$$ 4. For a repeated root, set the discriminant to zero: $$9 - 4k = 0.$$ 5. Solve for $k$: $$4k = 9$$ $$k = \frac{9}{4} = 2.25.$$ 6. Thus, the quadratic equation has one repeated root when $k = 2.25$. Final answer: $k = 2.25$