If the roots of ax² + 2bx + c = 0, a ≠ 0, a, b, c ∈ R are real and equal, then prove that a : b = b : c.

(To avoid confusion between variables here I have taken standard from as px² + qx + r = 0 and accordingly the discriminant)

Comparing equation ax² + 2bx + c = 0 with standard form px² + qx + r = 0 we get

p = a, q = 2b and r = c

Discriminant (D) = q² – 4pq

As roots are real and equal D = 0

⇒ q² – 4pq = 0

⇒ (2b)² – 4(a)(c) = 0

⇒ 4b² – 4ac = 0

⇒ 4b² = 4ac

⇒ b² = ac

⇒ b × b = a × c

⇒ =

⇒ b:c = a:b

⇒ a:b = b:c

Hence proved