If a, b, c R, a > 0, c < 0, then prove that the roots of ax² + bx + c = 0 are real and distinct.

For roots to be real and distinct D should be greater than 0

Discriminant (D) = b² – 4ac

In the discriminant b² is a positive number since square cannot be negative given a > 0 which means a is also positive

Now consider the product – 4 × c

As c < 0 c is negative

We are multiplying two negative numbers which would result in a positive number which means – 4 × c is also positive

So, we can conclude that b² – 4ac is positive

Hence D > 0

Hence roots are real and distinct