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## Main Question or Discussion Point

Let be the function h(x)=f(x)+g(x) we want to obtain the values of x so h(x)=0 and we have that f(x) and g(x) have the same roots (if a is so f(a)=0 then g(a)=0 too) so we have two types of roots:

1.-values of x that satisfy f(x)=-g(x)

2.values x that make f(x)=g(x)=0

then we make in the first equation f(x)/g(x)=-1 this should be valid for any root of h(x) the problem would be if the roots of f(x) and g(x) would also satisfy the formula f(a)/g(a)=-1 with a a root of f(x) and g(x) ,we have that the values that satisfy the equation h(x)=0->that f(x)/g(x)=-1 so as a is a root of f and g would also satisfy that h(a)=0=0+0 and so should satisfy that f(x)/g(x)=-1 but how to prove this?...thanks.

1.-values of x that satisfy f(x)=-g(x)

2.values x that make f(x)=g(x)=0

then we make in the first equation f(x)/g(x)=-1 this should be valid for any root of h(x) the problem would be if the roots of f(x) and g(x) would also satisfy the formula f(a)/g(a)=-1 with a a root of f(x) and g(x) ,we have that the values that satisfy the equation h(x)=0->that f(x)/g(x)=-1 so as a is a root of f and g would also satisfy that h(a)=0=0+0 and so should satisfy that f(x)/g(x)=-1 but how to prove this?...thanks.