If the quadratic equation $ax^2 + bx + 6 = 0 $ does not have distinct real roots, then the least value of $2a + b$ is

For no distinct real roots, the discriminant D = b^2 - 4ac <= 0. Here, b^2 - 4(a)(6) <= 0, so b^2 <= 24a. We want to minimize 2a + b. Let k = 2a + b, so b = k - 2a. Substituting into the inequality: (k - 2a)^2 <= 24a. For this to hold for some a, the quadratic in a must have a non-negative discriminant. Solving this leads to the minimum value -3.