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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
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Solve by factorization X 2 +(a/a+b + a+b/a)x+1 = 0 X 2 +(a/a+b + a+b/a)x+1 => X 2 +(a/a+b x a+b/ax + a/a+b .a+b/a) => X[x+a/a+b] +a+b/a[a+a*a+b]= 0 => X= -a
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