Not all quadratic equations will have two unique and real solutions. If the radicand in the quadratic formula is zero, there will be two equal solutions. If the radicand is negative, there will be no real solutions.

Quadratic Formula

The value of the radicand in the quadratic formula is consistent with what you would expect to see if you solved a quadratic equation graphically.

Example 2

Solve using the quadratic formula.

The two solutions are and .
This means the only solution to the equation is x = −3 and the graph of the corresponding quadratic function has its vertex lying on the x-axis.

Example 3

Solve using the quadratic formula.

Note that since the radicand is negative, there are no Real solutions to this equation.