What relationship exists between the zeros of the graph and the factored form of a quadratic equation?

The graph will have a formula such as y=x2+3x+2

This can be factorised in this example to y=(x+1)(x+2).

The zeroes of the graph are really the points which satisfy the simultaneous equations y=0 (a line) and y=(x+1)(x+2). Both of these are equal to y, so put them equal to each other: (x+1)(x+2)=0

This has only one variable, x, so we can find its values.

It is clear that if either x+1=0 or x+2=0 the whole expression will be equal to zero, as the bit that is zero will multiply the other bit by zero, so the whole thing is zero. Solve each of those two individually to get x=-1 and x=-2. These are the zeroes of the graph. Read it carefully. Many high school mathematics students were bored to bring you this information (I was bored at this stage but found it got much more interesting later).