Question

If roots of equation(a–b)x2 +(b–c)x+(c–a)=0are equal,prove that
2b=a+c.

Answer 1

I think you have typed wrong question :

correct question is :

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

your answer is here...

(b-c)x²+(c-a)x+(a-b)=0

Comparing with quadratic equation

Ax²+Bx+C=0

A=(b-c),

B=c-a,

C=a-b

Discriminate when roots are equal

D=B²-4AC=0

D=(c-a)²−4(b-c)(a-b)=0

D=(c²+a²−2ac)-4(ba-ac-b²+bc)=0

D=c²+a²−2ac-4ab+4ac+4b²-4bc=0

c²+a²+2ac-4b(a+c)+4b²=0

(a+c)²-4b(a+c)+4b²=0

[(a+c)-2b]²=0

a+c=2b

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