2、

Prove:

(1) The necessity of proof:

Because the quadratic function crosses the point (-1, 0), a-b+c=0, which proves the necessity;

(2) Proof of sufficiency:

B = a+c from a-b+c = 0;

ax^2+bx+c=0

ax^2+(a+c)x+c=0

ax^2+ax+cx+c=0

ax(x+ 1)+c(x+ 1)=0

(ax+c)*(x+ 1)=0

So ax+c=0 or x+ 1=0.

Get x=-c/a or x=- 1.

So the quadratic function crosses the point (-1, 0);

To sum up, the image of quadratic function y = ax 2+bx+c intersects the x axis (-1, 0) if and only if a-b+c=0.