If the roots of the equation are equal. Then:
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Let f ( x ) = ( a − b ) x 2 + ( b − c ) x + ( c − a ) . We see that f ( 1 ) = 0 . Therefore, since both roots are equal, f ( x ) = k ( x − 1 ) 2 = k x 2 − 2 k x + k . We see that the coefficient of x 2 and the constant term are equal.
Therefore, a − b = c − a ⟹ 2 a = b + c