if the roots of the equation (b-c)x^2+(c-a)x+(a-b)=0 are equal show that a,b,c are in AP

Dear Student

For the equation(b-c)x²+(c-a)x+(a-b)=0 to have equal roots Discriminant D must be zero

Let A = (b-c) B =(c-a), C = (a-b)

So B²-4AC = 0

(c-a)² =4(b-c)(a-b)

c²+a²-2ac =4ba-4b²-4ca+4bc

4b²+c²+a^2-2(2b)(c)-2(2b)(a)+2(a)(c)=0.

(c+a-2b)²=0

So 2b=c+a

or a,b,c are in AP