Graph of quadratic function y=ax^2+bx+c is a parabola, that is open upward if a>0, and open downward if a<0.

There are 3 possible cases:

1. parabola intersects x-axis (i.e. equation ax^2+bx+c=0 has two different roots);
2. parabola has vertex on x-axis (i.e. equation ax^2+bx+c=0 has one root);
3. parabola doesn't intersect x-axis (i.e. equation ax^2+bx+c=0 doesn't have roots).

Therefore, there are 6 possible positions of parabola, shown on figure.

Example 1. Solve graphically inequality 2x^2-5x+2>0.

Equation 2x^2-5x+2=0 has two roots: x_1=0.5,x_2=2. Parabola, that is graph of function y=2x^2-5x+2 has form like graph a) on the figure.

Inequality 2x^2-5x+2>0 holds for such x, that points of parabola lies above x-axis: this will be when x<x_1 or x>x_2, i.e x<0.5 or x>2.

Therefore, set of solutions of quadratic inequality is (-oo,0.5)uu(2,oo).

Example 2. Solve graphically inequality -x^2+4x-4>0.

Equation -x^2+4x-4=0 has one root x=2. Parabola, that is graph of function y=-x^2+4x-4 has form like graph e) on the figure.

Inequality 2x^2-5x+2>0 holds for such x, that points of parabola lies above x-axis: there are no such points, therefore inequality doesn't have solution.