If the roots of the equations and are simultaneously real, then prove that b² = ac.

Question

If the roots of the equations and are simultaneously real, then prove that b 2 = ac.

Philoid · Accepted Answer

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

D = b² – 4ac

If D ≥ 0, roots are real

⇒ 4b² – 4ac ≥ 0

⇒ b² ≥ ac ------ (1)

⇒ 4ac – 4b² ≥ 0

⇒ b² ≤ ac ----- (2)

For both (1) and (2) to be true

⇒ b² = ac

If the roots of the equations and are simultaneously real, then prove that b2 = ac.