For what values of k does the quadratic equation x ^ 2-3x + 12k = 0 have two roots?

We have an equation with a parameter:

x² – 3 * x + 12 * k = 0.

Because it is required that it have a pair of real roots, then its discriminant:

D = 9 – 48 * k> 0.

Let’s solve this linear inequality:

9 – 48 * k> 0.

Divide both parts by 48 and transfer k to the right side, we get:

9/48 – k> 0,

3/16> k, or k <3/16.

Therefore, k must take any values from the interval (-∞; 3/16), only then the original quadratic equation will have two valid solutions (roots).