let a, b, x be Real; :: thesis: ( a <> 0 & Polynom a,b,0 ,x = 0 & not x = 0 implies x = - (b / a) )
assume that
A1: a <> 0 and
A2: Polynom a,b,0 ,x = 0 ; :: thesis: ( x = 0 or x = - (b / a) )
((a * (x ^2 )) + (b * x)) + 0 = 0 by A2, POLYEQ_1:def 2;
then (((a * x) + b) + 0 ) * x = 0 ;
then ( ((a * x) + b) + (- b) = 0 + (- b) or x = 0 ) by XCMPLX_1:6;
then ( x = (- b) / a or x = 0 ) by A1, XCMPLX_1:90;
hence ( x = 0 or x = - (b / a) ) by XCMPLX_1:188; :: thesis: verum