The Quadratic Formula
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 equation ax^{2} + bx + c = 0, the solutions are given by the equation
x = 

Example 1: Solve for x: x^{2} + 8x + 15.75 = 0
a = 1, b = 8, and c = 15.75.
x =
=
=
=
= or
=  or
Thus,
x =  or
x =  .
Example 2: Solve for x: 3x^{2}  10x  25 = 0.
a = 3, b =  10, and c =  25.
x =
=
=
=
=
= or
= 5 or
Thus,
x = 5 or
x =  .
Example 3: Solve for x: 3x^{2}  24x  48 = 0.
a =  3, b =  24, and c =  48.
x =
=
=
=
=
= =  4
Thus,
x =  4.
Example 4: Solve for x: 2x^{2}  4x + 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
xaxis.)
The Discriminant
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}  4ac), is positive, negative, or zero. This expression has a special name: the discriminant.