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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 xintercepts when you have the function f (x) = ax^{2} + bx + c ,
you will have only one xintercept 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 xintercepts
when you have the function f (x) = ax^{2} + bx + c you will have no xintercepts
on the graph of the function.
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