Section 5.4
The Quadratic Formula
481
Version: Fall 2007
5.4
The Quadratic Formula
Consider the general quadratic function
f
(
x
) =
ax
2
+
bx
+
c.
In the previous section, we learned that we can find the zeros of this function by solving
the equation
f
(
x
) = 0
.
If we substitute
f
(
x
) =
ax
2
+
bx
+
c
, then the resulting equation
ax
2
+
bx
+
c
= 0
(1)
is called a
quadratic equation
. In the previous section, we solved equations of this type
by factoring and using the zero product property.
However, it is not always possible to factor the trinomial on the left-hand side of the
quadratic
equation (1)
as a product of factors with integer coefficients. For example,
consider the quadratic equation
2
x
2
+ 7
x
−
3 = 0
.
(2)
Comparing
2
x
2
+ 7
x
−
3
with
ax
2
+
bx
+
c
, let’s list all integer pairs whose product is
ac
= (2)(
−
3) =
−
6
.
1
,
−
6
−
1
,
6
2
,
−
3
−
2
,
3
Not a single one of these integer pairs adds to
b
= 7
.
Therefore, the quadratic
trinomial
2
x
2
+ 7
x
−
3
does not factor over the integers.
2
Consequently, we’ll need
another method to solve the quadratic
equation (2)
.
The purpose of this section is to develop a formula that will consistently provide
solutions of the general quadratic
equation (1)
. However, before we can develop the
“Quadratic Formula,” we need to lay some groundwork involving the square roots of
numbers.
Square Roots
We begin our discussion of square roots by investigating the solutions of the equation
x
2
=
a
. Consider the rather simple equation
x
2
= 25
.
(3)
Because
(
−
5)
2
= 25
and
(5)
2
= 25
,
equation (3)
has two solutions,
x
=
−
5
or
x
= 5
.
We usually denote these solutions simultaneously, using a “plus or minus” sign:
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1
This means that the trinomial
2
x
2
+ 7
x
−
3
cannot be expressed as a product of factors with integral
2
(integer) coefficients.

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482
Chapter 5
Quadratic Functions
Version: Fall 2007
x
=
±
5
These solutions are called
square roots
of 25. Because there are two solutions, we need
a different notation for each. We will denote the positive square root of 25 with the
notation
√
25
and the negative square root of 25 with the notation
−
√
25
. Thus,
√
25 = 5
and
−
√
25 =
−
5
.
In a similar vein, the equation
x
2
= 36
has two solutions,
x
=
±
√
36
, or alternatively,
x
=
±
6
. The notation
√
36
calls for the positive square root, while the notation
−
√
36
calls for the negative square root. That is,
√
36 = 6
and
−
√
36 =
−
6
.
It is not necessary that the right-hand side of the equation
x
2
=
a
be a “perfect
square.” For example, the equation
x
2
= 7
has solutions
x
=
±
√
7
.
(4)
There is no
rational
square root of 7. That is, there is no way to express the square
root of 7 in the form
p/q
, where
p
and
q
are integers. Therefore,
√
7
is an example of
an irrational number. However,
√
7
is a perfectly valid real number and we’re perfectly
comfortable leaving our answer in the form shown in
equation (4)
.

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