# Quadratic and Linear Function relationship

I just stumbled across this problem and would appreciate some help with it, as I'm not getting far with it.

Problem: You are given a quadratic function $$f(x)=ax^2+bx+c$$ and a linear function $$g(x)$$.

The two functions intersect at $$x=0$$ and at also at an $$x$$ with $$g(x)=f(x)=0$$ and where $$x<0$$.

Which of the two could, for some values of $$a,b,c$$, be an expression for $$g(x)$$:

1. $$g(x)=bx+c$$
2. $$g(x)=ax+c$$

My Progress thus far: We know that the $$y$$-intercept must be $$c$$, because $$f(x)$$ and $$g(x)$$ intercept at $$x=0$$. So that makes perfect sense. However I fail to see how for some values of $$a,b,c$$ the gradient of $$g(x)$$ could either be $$a$$ or $$b$$. Any help would greatly be appreciated.

Given f(x)=$$ax^2$$+bx+c ($$a \neq0$$),

if g(x)=bx+c for some x=$$x_1$$<0,

f($$x_1$$)=$${ax_1}^2$$+$$bx_1$$+c (I)

g($$x_1$$)=b$$x_1$$+c (II)

(I)-(II) $${ax_1}^2$$ = 0, a=0 , it is not allowed.

So, this is not the right choice, only option is (2) g(x)=ax+c.

You can prove that from here.