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Math 8H
8-4
Factoring Trinomials
ax2 + bx + c
Algebra 1
Glencoe McGraw-Hill
JoAnn Evans
Quadratic trinomials that have an
“a” value other than 1 require an
extra step beyond the factoring X.
●
+
ax2
c
To complete the process, you’ll
need to construct a 4-square box.
Factor:
a=2
-10
-5
●
+
-3
2
2x2 – 3x - 5
b = -3
c = -5
Put a∙c in the top of the X.
2 ∙ -5 = -10
Put b in the bottom of the X.
Name two numbers that have a
product of -10 and a sum of -3.
2x2 – 3x - 5
-10
-5
●
+
-3
2
2x2
2x
-5x
-5
Put ax2 in the
top left corner.
Put c in the lower
right corner.
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.
Notice that all terms of the trinomial are represented in the
box. The ax2 and the c appear as they do in the original
trinomial. The middle term of the original trinomial is the sum
of the remaining squares.
2x2 – 3x - 5
x + 1
2x 2x2
2x
5 -5x
-5
-
(2x – 5)(x + 1)
is the factored form.
Find the GCMF in each column.
Write it above the column.
Always pull out only the positive
common factors.
Find the GCMF in each row.
Write it beside the row.
Always pull out only the positive
common factors.
The sign between the two terms will be the sign
of the coefficient in the diagonal boxes.
6x2 + 11x + 4
3x + 4
24
3
●
+
11
8
2x 6x2
+
1
3x
Find the GCMF in each
row and each column.
(3x + 4)(2x + 1)
is the factored form.
Check your
solution!
8x
4
Put ax2 in the
top left corner.
Put c in the lower
right corner.
Put the side numbers
in the two remaining
boxes with an “x”
after them. Their
order is not
important.
5x2 - 28x - 12
5x + 2
-60
●
-30 + 2
-28
x
-
5x2
2x
6 -30x -12
Find the GCMF in each
row and each column.
(5x + 2)(x - 6)
is the factored form.
Check your
solution!
Put ax2 in the
top left corner.
Put c in the lower
right corner.
Put the side numbers
in the two remaining
boxes with an “x”
after them. Their
order is not
important.
14x2 - 5x - 1
7x + 1
-14
-7
●
+
-5
2
2x 14x2 2x
1
-7x
Find the GCMF in each
row and each column.
(7x + 1)(2x - 1)
is the factored form.
Check your
solution!
-1
Put ax2 in the
top left corner.
Put c in the lower
right corner.
Put the side numbers
in the two remaining
boxes with an “x”
after them. Their
order is not
important.
5x2 + 13x + 6
5x + 3
30
10
●
+
13
3
x
+
5x2 3x
2 10x
Find the GCMF in each
row and each column.
(5x + 3)(x + 2)
is the factored form.
Check your
solution!
6
Put ax2 in the
top left corner.
Put c in the lower
right corner.
Put the side numbers
in the two remaining
boxes with an “x”
after them. Their
order is not
important.
On your own:
6x2 + x - 12
2x2 - 7x - 9
-72
9
●
+
-18
●
-9 + 2
-8
1
-7
2x + 3
x + 1
3x 6x2 9x
2x 2x2 2x
-
4 -8x
-12
(2x + 3)(3x - 4)
-
9 -9x
-9
(x + 1)(2x - 9)
Caution! This method will NOT work if
the trinomial has a GCMF that could be
factored out first.
Example: 6x2 – 8x - 8
-12
●
-6 + 2
-4
3x + 2
x
2
3x2
-6x
2x
-4
Factor out the GCMF of 2
before proceeding with
the X-Box method!
2(3x2 – 4x – 4)
a=3
b = -4
c = -4
2(3x + 2)(x - 2)
is the factored form.
Don’t forget to include the GCMF
that was factored out first!
Caution! Look for a GCMF that can be
factored out first!
2x2y + 10xy – 72y
2y(x2 + 5x – 36)
-36
●
-4 + 9
5
What is the GCMF of
these three terms?
2y
2y(x2 + 5x – 36)
a=1
b=5
c = -36
2y(x - 4)(x + 9)
is the factored form.
Don’t forget to include the GCMF that was
factored out first!
If there are 4 or more terms,
factor by the grouping method.
10x2
- 8x - 35x + 28
The first two terms have
a common factor of 2x.
2x(5x – 4) -7(5x – 4)
What common factor do
the second two terms
have? -7
(2x – 7)(5x – 4)
Factor out the GCMF
from each pair of terms.
Write the expression
in factored form.
If there are 4 or more terms,
factor by the grouping method.
3x3 - 5x2 + 9x - 15
x2(3x – 5) + 3(3x – 5)
(x2
+ 3)(3x – 5)
The first two terms have
a common factor of x2.
What common factor do
the second two terms
have? 3
Factor out the GCMF
from each pair of terms.
Write the expression
in factored form.
Solve Equations by Factoring
3x2 – 5x = 12
Collect all terms on the left
hand side of the equation.
-36
●
-9 + 4
-5
3x2 – 5x – 12 = 0
3x + 4
Factor the left side using the
X-Box method.
(3x + 4)(x – 3) = 0
Use the Zero Product Property.
3x + 4 = 0
or x – 3 = 0
4
or 3
x= 
3
x
3
3x2
-9x
4x
-12