If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations and has real roots.

Question

If a, b, c are real numbers such that ac &#x2260; 0, then show that at least one of the equations and has real roots.

Philoid · Accepted Answer

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

D = b² – 4ac

If D > 0, roots are real.

⇒ D = b² – 4ac

If D > 0 then b² > 4ac -------- (1)

⇒ 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.