If the roots of the equation (b-c)x^2 + (c-a)x +(a-b) =0 are equal then prove that 2b = a+c

If the roots of the equation (b-c)x^2 + (c-a)x +(a-b) =0 are equal the

1.	If the roots of the equation (b-c)x^2 + (c-a)x +(a-b) =0 are equal then prove that 2b = a+c
Answer» {tex} (b-c)x^2 + (c-a)x +(a-b) =0{/tex}roots of te given equation are equal so\xa0{tex}{b^2-4ac}=0{/tex}{tex}\\implies (c-a)^2-4 (b-c)(a-b)=0{/tex}{tex}\\implies c^2-2ac+a^2-4(ab-b^2-ac+bc)=0{/tex}{tex}\\implies c^2-2ac+a^2-4ab+4b^2+4ac-4bc=0{/tex}{tex}\\implies a^2+4b^2+c^2-4ab-4bc+2ac=0{/tex}{tex}\\implies (a-2b+c)^2=0{/tex}{tex}\\implies (a-2b+c)=0{/tex}{tex}\\implies a+c=2b{/tex}{tex}2b= a+c\\; \\; Hence\\; proved{/tex}\xa0