Show the student some graphs of quadratics. It soon becomes obvious that the graph of a quadratic is a parabola and there are three ways that a parabola can interact with the X axis: not at all; touches in one place only; crosses the X axis in two places. If the parabola touches the X axis at one point only, the two real roots are equal and the quadratic is a perfect square. If the parabola crosses the X axis at two points, there are two real roots and two factors that are different.
Explain that when you are given the solutions you are given the points where the parabola interacts with the X axis. This makes sense because the solutions are where Y = 0 and all of these points are on the X axis. The relationship between the quadratic, the factors and the roots is this: The solutions of the quadratic are the same as the solutions to all of the factors. This means that if the parabola crosses the X axis at point p, this is not only a solution of the quadratic but a solution of one of the factors of the quadratic. A logical choice for this factor is X - p, because the solution of X - p = 0 is X = p.
Show the student that when you are given the solutions p and q, the factors are X - p and X - q and the quadratic is the product (X - p)(X - q). You should point out that if the solution 3/4, the corresponding factor could be X - 3/4, but it is more likely to be 4X - 3.