1. **State the problem:** We are given the quadratic equation $$kx^2 - 4x + 36k = 0$$ and told it has two equal roots with $$k > 0$$. We need to find the value of $$k$$. 2. **Recall the condition for equal roots:** A quadratic equation $$ax^2 + bx + c = 0$$ has two equal roots if its discriminant $$\Delta = b^2 - 4ac = 0$$. 3. **Identify coefficients:** Here, $$a = k$$, $$b = -4$$, and $$c = 36k$$. 4. **Write the discriminant condition:** $$\Delta = (-4)^2 - 4 \times k \times 36k = 0$$ 5. **Simplify the discriminant:** $$16 - 144k^2 = 0$$ 6. **Solve for $$k$$:** $$144k^2 = 16$$ $$k^2 = \frac{16}{144} = \frac{1}{9}$$ 7. **Take the positive root (since $$k > 0$$):** $$k = \frac{1}{3}$$ **Final answer:** $$k = \frac{1}{3}$$