1.3
Quadratic Equations
121
EXAMPLE 8
Completing the Square When the Leading
Coefficient Is Not Equal to
1
Solve the equation 3
x
2
12
x
13
0 by completing the square.
Solution:
Divide by the leading coefficient, 3.
Collect variables to one side of the equation
and constants to the other side.
Add
to both sides.
Write the left side of the equation as a
perfect square and simplify the right side.
Solve using the square root method.
Simplify.
Rationalize the denominator (Chapter 0).
Simplify.
,
The solution set is
.
■
YOUR TURN
Solve the equation 2
x
2
4
x
3
0 by completing the square.
e
2

i
1
3
3
, 2
+
i
1
3
3
f
x
=
2
+
i
1
3
3
x
=
2

i
1
3
3
x
=
2
;
i
1
3
#
1
3
1
3
x
=
2
;
i
A
1
3
x

2
=
;
A

1
3
(
x

2)
2
= 
1
3
x
2

4
x
+
4
= 
13
3
+
4
A

4
2
B
2
=
4
x
2

4
x
= 
13
3
x
2

4
x
+
13
3
=
0
EXAMPLE 7
Completing the Square
Solve the quadratic equation
x
2
8
x
3
0 by completing the square.
Solution:
Add 3 to both sides.
x
2
8
x
3
Add
to both sides.
x
2
8
x
4
2
3
4
2
Write the left side as a perfect square
and simplify the right side.
(
x
4)
2
19
Apply the square root method to solve.
Subtract 4 from both sides.
The solution set is
4
,
4
.
In Example 7, the leading coefficient (the coefficient of the
x
2
term) is 1. When the leading
coefficient is not 1, start by first dividing the equation by that leading coefficient.
F
1
19
1
19
E
x
= 
4
;
1
19
x
+
4
= ;
1
19
A
1
2
#
8
B
2
=
4
2
Technology Tip
Graph
y
1
x
2
8
x
3.
Study Tip
When the leading coefficient is
not 1, start by first dividing the
equation by that leading coefficient.
Technology Tip
Graph
y
1
3
x
2
12
x
13.
■
Answer:
The solution is
. The solution set is
e
1

i
1
2
2
, 1
+
i
1
2
2
f
.
x
=
1
;
i
2
2
2
The
x
intercepts are the solutions to
this equation.
The graph does not cross the
x
axis,
so there is no real solution to this
equation.
122
CHAPTER 1
Equations and Inequalities
Study Tip
Read as “negative
b
plus or minus
the square root of the quantity
b
squared minus 4
ac
all over 2
a
.”
x
=

b
;
2
b
2

4
ac
2
a
If
ax
2
bx
c
0,
, then the solution is
Note:
The quadratic equation must be in standard form (
ax
2
bx
c
0) in order
to identify the parameters:
a
—coefficient of
x
2
b
—coefficient of
x
c
—constant
x
=

b
;
2
b
2

4
ac
2
a
a
Z
0
Q
UADRATIC FORMULA
We read this formula as
negative b plus or minus the square root of the quantity b squared
minus 4ac all over 2a.
It is important to note that negative
b
could be positive (if
b
is
negative). For this reason, an alternate form is “opposite
b
. . .” The Quadratic Formula
should be memorized and used when simpler methods (factoring and the square root
method) cannot be used. The Quadratic Formula works for
any
quadratic equation.
Study Tip
The Quadratic Formula works for
any
quadratic equation.
Quadratic Formula
Let us now consider the most general quadratic equation:
We can solve this equation by completing the square.
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 Summer '17
 juan alberto
 Quadratic Formula, Quadratic equation, Equations and Inequalities