Solving Quadratic Inequalities Graphically

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:graphical solution of quadratic equation

  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.