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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.

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