# The Discriminant

The discriminant is a very useful concept when working
with quadratic functions . It can tell you about the

number and types of solution you will have to a quadratic equation. Similarly,
it can tell you about the

number of x - intercepts you will have on the graph of the quadratic equation. The
discriminant is the part

of the quadratic formula that is underneath the radical, and thus is equal to
b^2 - 4ac . Since quadratic

equations can have 0, 1, or 2 solutions, there are three different situations
that can occur in quadratic

equations and the graphs of their related functions. Given the quadratic
equation 0 = ax^2 + bx + c

In this situation, since b^2 - 4ac is a positive number ,
our quadratic formula

will simplify to And since the square root of

a positive number is another positive number,

Here you will have two real solutions:

one being

the other being

Similarly, since the quadratic equation 0 = ax^{2} + bx + c
is what you use to

find your x intercepts when you have the function f (x) = ax^{2} + bx + c ,

you will also have two x -intercepts on the graph of your function.

In this situation, since b^2 - 4ac is equal to zero, our
quadratic formula will

simplify to And since the square root of zero
is zero ,

This is just one real
solution for x .

Similarly, since the quadratic equation 0 = ax^{2} + bx + c is what you use to

find your x-intercepts when you have the function f (x) = ax^{2} + bx + c ,

you will have only one x-intercept on the graph of the function.

In this situation, since b^2 - 4ac is a negative number ,
our quadratic formula

will simplify to But, we know that taking

the square root of a negative number yields an imaginary number (i), and

not a real number , we will get two imaginary solutions. We could also

state that we have zero real solutions to this equation. Again, similarly, since
the

quadratic equation 0 = ax^{2} + bx + c is what you use to find your x-intercepts

when you have the function f (x) = ax^{2} + bx + c you will have no x-intercepts

on the graph of the function.

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