Question

if the roots of the equeation (a-b)x2 +(b-c)x + (c-a)=0 are equal,prove that 2a=b+c plz solve dis

Answer 1

Given quadratic equation is (a – b)x² + (b – c)x + (c – a) = 0
Since the root are equal, discriminent of the quadritic equation = 0
Hence (b – c)² = 4(a – b)(c – a)
⇒ b² + c² – 2bc = 4(ac – a² – bc + ab)
⇒ b² + c² – 2bc = 4ac – 4a2 – 4bc + 4ab
⇒ b² + c² – 2bc – 4ac + 4a² + 4bc – 4ab = 0
⇒ b² + c²+ 4a² + 2bc – 4ac – 4ab = 0
⇒ b² + c² + (2a)² + 2(b)(c) – 2(2a)c – 2(2a)b = 0
⇒ b² + c² + (–2a)² + 2(b)(c) + 2(–2a)c + 2(–2a)b = 0
⇒ (b + c – 2a)² = 0
⇒ b + c – 2a = 0
∴ b + c = 2a