Real Roots and Distinct Roots

Question

Show that the roots of the equation kx^2 + 5x - 2k = 0 are real and distinct for all the real values of k. How do I solve this? Because if I use the Quadratic formula then at one point there is the square root of a negative value which makes it undefined. So how do i prove or show them?

Answer 1

The discriminant is b² - 4ac, and for the roots to be real, it must be at least 0.

In this problem, a=k, b=5, and c=-2k. Hence, the discriminant is

5² - 4(k)(-2k) = 25 + 8k²

which is positive for all real k.