Trinomials are not always easy to factor. In fact, some trinomials cannot be factored. Thus, we need a different way to solve quadratic equations. Herein lies the importance of the quadratic formula:

Given a quadratic equationax^{2}+bx+c= 0 , the solutions are given by the equation

x=

*Example 1*: Solve for
*x*
:
*x*
^{2} + 8*x* + 15.75 = 0

*a* = 1
,
*b* = 8
, and
*c* = 15.75
.

*x* =

=Thus,

=

=

=or

= -or-

*Example 2*: Solve for
*x*
:
3*x*
^{2} - 10*x* - 25 = 0
.

*a* = 3
,
*b* = - 10
, and
*c* = - 25
.

*x* =

=Thus,

=

=

=

=or

= 5or-

*Example 3*: Solve for
*x*
:
-3*x*
^{2} - 24*x* - 48 = 0
.

*a* = - 3
,
*b* = - 24
, and
*c* = - 48
.

*x* =

=Thus,

=

=

=

= = - 4

*Example 4*: Solve for
*x*
:
2*x*
^{2} - 4*x* + 7
.

*a* = 2
,
*b* = - 4
, and
*c* = 7
.

*x* =

=Since we cannot take the square root of a negative number, there are no solutions. (The graph of this quadratic polynomial will therefore be a parabola that never touches the

=

=

As we have seen, there can be
0
,
1
, or
2
solutions to a quadratic equation, depending on whether the expression inside the square root sign,
(*b*
^{2} - 4*ac*)
, is positive, negative, or zero. This expression has a special name: the discriminant.