Common zero of two quadratics

Question

Consider$ f(x) =x^2 + ax+b$ , and $g(x) = x^2 +bx+a$ , given that both have one common zero, what is the value of a+b, given $ a\neq b$

Solution according to book:

f(1)= g(1)=0 and, hence, a+b=-1

But this doesn't make sense to me, as they could intersect at other points too. Like how would u know that the 0 is the only point where the curves intersect?

Answer 1

Let $x_0$ be a point where the two functions intersect. This implies $f(x_0)= g(x_0)$. On plugging in the values, you can see that there is only one possible solution for that equation (which will turn out to be $x_0 = 1$). Furthermore, since it is given that they have a common zero, it means the point of intersection is a zero which implies $f(1) = g(1) = 0$.

Answer 2

By subtraction,

$$(a-b)x+(b-a)=0$$ and $x=1$ is a solution.