Why Does p+q Equal Zero in a Special Quadratic Equation?

[requal]

1. Consider the quadratic equation ax^2+px+aq+q=0 where a does not equal to zero & p and q are constants . It is known that one of the roots of the quadratic equation is always 1 regardless of the value of a. Prove that p+q=0

3. I have tried to factorize it and it became x(ax+p)+q(a+1)=0 but that's it. I kinda stuck /b]

 

[Unco]

Hi Requal,

1. Consider the quadratic equation ax^2+px+aq+q=0 where a does not equal to zero & p and q are constants . It is known that one of the roots of the quadratic equation is always 1 regardless of the value of a. Prove that p+q=0

That 1 is a root of the quadratic equation ax^2 + px + aq + q = 0 means that x=1 satisfies the equation; so plugging in x=1 we have a + p + aq + q = 0.
We are told this is true for any (nonzero) value of a. Can you take it from here?