Question

Given \( a + b + c = 0 \) and \( a \neq c \), and \( ax^2 + bx + c = 0 \), what will be the nature of the roots?

Hint:

For quadratic equations, check the discriminant to determine the nature of the roots.

Answer 1

Step 1: Use the condition \( a + b + c = 0 \).
Substitute \( b = -(a+c) \) into the discriminant: \[ \Delta = b^2 - 4ac = (a+c)^2 - 4ac = (a-c)^2 \] Since the discriminant is a perfect square, the roots are real and rational.
Step 2: Roots are rational and one root is 1.
By solving the quadratic equation, we find that one root is 1, and the other is rational.