2a+3b+6c=0(a,b,c$\epsilon $ R) then the quadratic equation ${ ax }^{ 2 }+bx+c=0$

Let f of x equal a x squared plus b x plus c. Since two a plus three b plus six c equals zero, dividing by six gives one third a plus one half b plus c equals zero, which means f of one equals zero. The expression one third a plus one half b plus c also perfectly matches f of zero plus f of one half plus f of one divided by three. If the average of f of zero, f of one half, and f of one is zero, at least one of these values must be less than or equal to zero, ensuring a root exists in the interval from zero to one by the Intermediate Value Theorem.

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