Math 8H

1
Math 8H
8-4 Factoring Trinomials ax2 bx c
Algebra 1 Glencoe McGraw-Hill
JoAnn Evans
2
Quadratic trinomials that have an a value other
than 1 require an extra step beyond the factoring
X.
ax2
?

c
To complete the process, youll need to
construct a 4-square box.
3
Factor 2x2 3x - 5
a 2 b -3 c -5
-10
?
-5
2

Put ac in the top of the X. 2 -5 -10
-3
Put b in the bottom of the X.
Name two numbers that have a product of -10 and a
sum of -3.
4
2x2 3x - 5
Put ax2 in the top left corner.
-10
2x2
2x
?
-5
2

Put c in the lower right corner.
-5
-5x
-3
Put the numbers from the sides of the X figure in
the two remaining boxes. Their order is not
important. Add an x after each of the side
numbers.
What do you notice about the cross product in the
box?
The cross products are equal.
5
(2x 5)(x 1) is the factored form.
2x2 3x - 5
Find the GCMF in each column. Write it above the
column. Always pull out only the positive
common factors.

x
1
2x2
2x
2x
-
Find the GCMF in each row. Write it beside the
row. Always pull out only the positive
common factors.
-5
-5x
5
The sign between the two terms will be the sign
of the coefficient in the diagonal boxes.
6
6x2 11x 4

3x
4
Put ax2 in the top left corner.
24
6x2
8x
2x
?
3
8


Put c in the lower right corner.
11
4
3x
1
Put the side numbers in the two remaining boxes
with an x after them. Their order is not
important.
Find the GCMF in each row and each column.
(3x 4)(2x 1) is the factored form.
Check your solution!
Are the cross products equal?
7
5x2 - 28x - 12

5x
2
Put ax2 in the top left corner.
-60
5x2
2x
x
?
-30
2
-

Put c in the lower right corner.
-28
-12
-30x
6
Put the side numbers in the two remaining boxes
with an x after them. Their order is not
important.
Find the GCMF in each row and each column.
(5x 2)(x - 6) is the factored form.
Check your solution!
Are the cross products equal?
8
14x2 - 5x - 1

7x
1
Put ax2 in the top left corner.
-14
14x2
2x
2x
?
-7
2
-

Put c in the lower right corner.
-5
-1
-7x
1
Put the side numbers in the two remaining boxes
with an x after them. Their order is not
important.
Find the GCMF in each row and each column.
(7x 1)(2x - 1) is the factored form.
Check your solution!
Are the cross products equal?
9
5x2 13x 6

5x
3
Put ax2 in the top left corner.
30
5x2
3x
x
?
10
3


Put c in the lower right corner.
13
6
10x
2
Put the side numbers in the two remaining boxes
with an x after them. Their order is not
important.
Find the GCMF in each row and each column.
(5x 3)(x 2) is the factored form.
Check your solution!
Are the cross products equal?
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Practice Problems
6x2 x - 12
2x2 - 7x - 9
-72
-18
?
?
2
-9
-8
9


1
-7


2x
3
x
1
2x
6x2
9x
2x2
2x
3x
-
-
-12
-8x
4
-9
-9x
9
(2x 3)(3x - 4)
(x 1)(2x - 9)
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Caution! This method will NOT work if the
trinomial has a GCMF that could be factored out
first.
Factor out the GCMF of 2 before proceeding with
the X-Box method!
Example 6x2 8x - 8
-12
?
-6
2
2(3x2 4x 4)

-4
a 3 b -4 c -4
3x
2

2(3x 2)(x - 2) is the factored form.
3x2
2x
x
-
-4
-6x
2
Dont forget to include the GCMF that was
factored out first!
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If there are 4 or more terms, factor by
the grouping method.
The first two terms have a common factor of 2x.
What common factor do the second two terms have?

10x2 - 8x - 35x 28
2x(5x 4)
-7(5x 4)
-7
Factor out the GCMF from each pair of terms.
(2x 7)(5x 4)
Write the expression in factored form.
13
If there are 4 or more terms, factor by
the grouping method.
The first two terms have a common factor of x2.
What common factor do the second two terms have?

3x3 - 5x2 9x - 15
x2(3x 5) 3(3x 5)
3
Factor out the GCMF from each pair of terms.
(x2 3)(3x 5)
Write the expression in factored form.
14
Solve Equations by Factoring
-36
3x2 5x 12
?
-9
4
Collect all terms on the left hand side of the
equation.

-5
3x2 5x 12 0
3x
4

Factor the left side using the X-Box method.
3x2
4x
x
-
(3x 4)(x 3) 0
-12
-9x
3
Use the Zero Product Property.
3x 4 0 or x 3 0
x