If the roots of the equation (b - c)x^2 + (c - a)x + (a - b) = 0 are equal, then prove that 2b = a + c.

Question

If the roots of the equation \((b-c)\text{x}^2+(c-a)\text{x}+(a-b)=0\) are equal, then prove that 2b = a + c.

Answer 1

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

D = b² – 4ac

If D = 0, roots are equal

\((b-c)\text{x}^2+(c-a)\text{x}+(a-b)=0\)

⇒ (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