If the roots of the equation are equal, then prove that 2b = a + c.

Question

Philoid · Accepted Answer

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

D = b² – 4ac

If D = 0, roots are equal

⇒ (c – a)² – 4(b – c)(a – b) = 0

⇒ c² + a² – 2ac + 4b² – 4ab - 4cb + 4ac = 0

⇒ a² + 4b² + c² + 2ac – 4ab – 4bc = 0

⇒ (a – 2b + c)² = 0

⇒ 2b = a + c