Question

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

Answer 1

Clearly, x=1 satisfies the given equation.
Since its roots are equal, so 1 and 1 are its roots.
`:." ""product of roots of the given equation "=(1xx1)=1.`
But, product of roots `=(a-b)/(b-c)." "[because" product of roots "=(C)/(A)]`
`:." "(a-b)/(b-c)=1impliesa-b=b-cimplies2b=a+c.`
Hence, `2b=a+c.`