If the roots of quadratio equation (a-b)x×x+(b-c)X+(c-a)=0 are equal .prove that 2a=b+c

If the roots of quadratio equation (a-b)x×x+(b-c)X+(c-a)=0 are equal

1.	If the roots of quadratio equation (a-b)x×x+(b-c)X+(c-a)=0 are equal .prove that 2a=b+c
Answer» We have (a-b)x2 + (b-c)x + (c-a) = 0Here A\xa0= (a-b), B\xa0= (b-c), C= (c-a){tex}\\therefore D = B^2 - 4AC = (b-c)^2 - 4(a-b) (c-a){/tex}For equal roots, D = 0{tex}\\implies{/tex}(b-c)2 - 4 (a-b) (c-a) = 0b2+c2-2bc -4(ac-a2-bc+ab) =0b2+c2-2bc -4ac+4a2+4bc-4ab=04a2+b2+c2+2bc-4ab-4ac=0(2a-b-c)2=0i.e. 2a-b-c =0Hence, b + c = 2a or 2a = b + c