if the roots of the equation a-bx2 b-c x c - a 0

Question

If the roots of the equation (a-b)x² + (b-c) x+ (c - a)= 0 are equal. Prove that 2a=b+c.

Ans:    (a-b)x² + (b-c) x+ (c - a) = 0

T.P 2a = b + c

B² - 4AC = 0

(b-c)² - [4(a-b) (c - a)] = 0

b²-2bc + c² - [4(ac-a² - bc + ab)] = 0

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

⇒ b²+ 2bc + c² + 4a2 - 4ac - 4ab= 0

⇒ (b + c - 2a)² = 0

⇒ b + c  = 2a