THere are three general cases of the quadratic equation

Case 1 all variables are present

In this case you can attempt to factor or utilize the quadratic formula

This in its self leaves three possiblity

if There are two real solutions

if there is one real solution

if there are no real solutions this arises from the domain conflict of when this is negative

Case 2 c=0

This yields

Factoring we get

Expanding upon that utilizing the zero-product property we get

or

Case 3 b=0

This gives

solving for x we get

Realizing that we must stipulate that to have a real solution.

This should provide all the information on quadratics you will need

As for graphing inequalities look here Graphing Linear Inequalities