Explanation:  The Fundamental Theorem of Algebra tells us that a quadratic function (i.e. a degree two polynomial) will always have two roots.  As we know from experience however, not every quadratic function has two x-intercepts.  We can find out how many x-intercepts a graph has by looking at its roots.  There will be a x-intercept for every unique real root of a quadratic function.  Complex roots will not give us x-intercepts but any real root (integer, rational, or irrational) will.  There are only three different cases; two x-intercepts, one x-intercepts or no x-intercepts.

• Case 1 - Two different real roots...   gives us two x-intercepts
• Case 2 - Two identical real roots...   gives us one x-intercept
• Case 3 - Two complex roots...   gives us no x-intercepts