If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax^2 + bx + c = 0 and -ax^2 + bx + c = 0 has real roots.

Question

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations \(a\text{x}^2+b\text{x}+c=0\) and -ax² + bx + c = 0 has real roots.

Answer 1

For a quadratic equation, ax² + bx + c = 0, 

D = b² – 4ac 

If D > 0, roots are real.

ax² + bx + c = 0

⇒ D = b² – 4ac 

If D > 0 then b² > 4ac -------- (1)

-ax² + bx + c = 0

⇒ D = b² + 4ac 

If D > 0, b² > - 4ac --------- (2) 

If (1) is true then (2) is false and vice versa 

Thus, one of the equation has real roots.