1. **Problem statement:** Determine the condition on $k$ such that the quadratic equation $kx^2 + 3x + 6 = 0$ has two unique real roots. 2. **Formula used:** For a quadratic equation $ax^2 + bx + c = 0$, the discriminant $\Delta$ is given by: $$\Delta = b^2 - 4ac$$ 3. **Important rule:** - If $\Delta > 0$, the equation has two distinct real roots. - If $\Delta = 0$, the equation has exactly one real root (a repeated root). - If $\Delta < 0$, the equation has no real roots (complex roots). 4. **Apply to given equation:** Here, $a = k$, $b = 3$, and $c = 6$. Calculate the discriminant: $$\Delta = 3^2 - 4 \times k \times 6 = 9 - 24k$$ 5. **Condition for two unique real roots:** $$\Delta > 0 \implies 9 - 24k > 0$$ 6. **Solve inequality:** $$9 > 24k$$ $$k < \frac{9}{24} = \frac{3}{8}$$ 7. **Final answer:** The quadratic equation $kx^2 + 3x + 6 = 0$ has two unique real roots if and only if $$k < \frac{3}{8}$$