1. The problem states that the roots of the equation $f(x) = 0$ are given by $$x = \frac{-5 \pm \sqrt{3 - 12k^2}}{4}.$$ We need to find the values of $k$ for which the roots are equal. 2. Roots are equal when the discriminant (the expression inside the square root) is zero. So, set the discriminant equal to zero: $$3 - 12k^2 = 0.$$ 3. Solve for $k^2$: $$12k^2 = 3$$ $$k^2 = \frac{3}{12} = \frac{1}{4}.$$ 4. Take the square root of both sides to find $k$: $$k = \pm \frac{1}{2}.$$ 5. Therefore, the roots are equal when $k = \frac{1}{2}$ or $k = -\frac{1}{2}$. Final answer: $$k = \pm \frac{1}{2}.$$