The Discriminant

The quadratic formula,
_{},
can be a useful tool in determining the solution(s) to a quadratic equation.
The nature of the solution(s) can be determined by a quick look at the *
discriminant. *The radicand of the
quadratic formula is the *
discriminant*.
In other words the *
discriminant* is b^{2 }- 4ac.

Using the standard form of the quadratic equation
which is
_{},
the nature of the solution(s) to any quadratic formula can then be
determined by checking the *
discriminant*.

(1) If b^{2 }- 4ac = 0, then the quadratic
equation will have exactly *one Real number
solution* (with duplicity.)
In other words, there is one solution that is repeated. This
occurs in the case of a perfect square trinomial such as x^{2 }-
6x + 9 whose *
discriminant *would be (-6)^{2 }- 4(1)(9) = 0.
When factored you would normally arrive at the solution (x -3)(x -3) = 0
or x = 3 and x =3, thus the "duplicity" of the answer x = 3.

(2) If b^{2 }- 4ac > 0 (in other words the
*
discriminant *is a positive number), then the quadratic equation will have
exactly *two Real number Solutions*.
For example consider, x^{2 }- 2x -8 whose *
discriminant* would be
(-2)^{2 }- 4(1)(-8) = 36. Since 36 is greater than 0,
there are two (unequal) Real solutions.

(3) If b^{2 }- 4ac < 0 (in other words the
*
discriminant* is a negative number), then the quadratic equation will have
no Real solutions. Instead there will be*
two* *C**omplex
number* solutions. For example consider, x^{2 }- 3x +
6 whose *
discriminant* would be (-3)^{2 }- 4(1)(6) = -15.
Since -15 is less than 0, there are two
Complex Number (conjugates) solutions.