if alpha is S......of ax^2+bx+c=0

then a(alpha^2)+b(alpha)+c=0

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Not sure just what "alpha is S......" means. Is that supposed to be an integral?

But, since x is just a place holder, you know that if you replace x with alpha, then ax^2+bx+c=0 just becomes

a(alpha^2)+b(alpha)+c=0

which is certainly true for any solution of the equation.