The Discriminant

The solutions of a quadratic equationare given by

In order for the equation to have real solutions, we must have thatsince we cannot take the square root of a negative number to obtain a real number. This means that if the equation has no real solutions. Ifthere are solutions, given byObviously these two solutions are the same

If b^2 -4ac >0 then the solutions are distinct, given byand

The expressionis called the discriminant and the number of solutions of a quadratic is determined solely by whether it is positive, negative, or zero. It is often labelledThe number of roots, and their significance on the graph of the quadratic, is illustrated in the graph below.

Notice that if there are no roots, the graph doesn't cross the x – axis at all.

If there is one root, the graph touches the x – axis (at one point) but doesn't cross it.

If there are two roots, the graph crosses the x – axis in two separate places.

Suppose thatthe discriminant isso the equation has no solutions.