Notes 8-7: Factoring and
Creating Perfect Square
Trinomials
I. Review from Unit 10
Square of a Sum
Square of a Difference
II. Perfect Square Trinomials
• Remember that the Square of a Sum and the Square
of a Difference result in a Perfect Square Trinomial
(PST). Recognizing PSTs and working backwards
from the ‘formula’ will allow you to factor much
faster than using x method, grouping, or box
method. For some PSTs, it will be the ONLY
factoring method that will work.
1) Factor x2 + 6x + 9
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = x
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(x + 3)2
Yes, b = 3
3) Is the middle term twice the
You can still
product of the a and b?
factor the other way
but this is quicker!
Yes, 2ab = 2(x)(3) = 6x
2) Factor y2 – 16y + 64
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = y
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(y – 8)2
Yes, b = 8
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(y)(8) = 16y
3) Factor m2 – 12m + 36
1.
2.
3.
4.
(m – 6)(m + 6)
(m – 6)2
(m + 6)2
(m – 18)2
4) Factor 4p2 + 4p + 1
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 2p
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(2p + 1)2
Yes, b = 1
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(2p)(1) = 4p
5) Factor 25x2 – 110xy + 121y2
Does this fit the form of our
Perfect Square Trinomials
perfect square trinomial?
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 5x
Since all three are true,
2) Is the last term a perfect
write your answer!
square?
(5x – 11y)2
Yes, b = 11y
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(5x)(11y) = 110xy
6) Factor 9k2 + 12k + 4
1.
2.
3.
4.
(3k + 2)2
(3k – 2)2
(3k + 2)(3k – 2)
I’ve got no
clue…I’m lost!
7) Factor 2r2 + 12r + 18
1.
2.
3.
4.
5.
prime
2(r2 + 6r + 9)
2(r – 3)2
2(r + 3)2
2(r – 3)(r + 3)
Don’t forget to factor the
GCF first!
III. Creating PSTs by Finding ‘c’
• For future tasks (completing the square,
etc), you will need to create PSTs.
Remember that standard form of a quadratic
equation is ax2+bx+c. You will often be
given the first two terms, and you will have
to find a value for ‘c’ that will create a PST.
Rule for Creating PSTs
ax  bx
2
b
x  bx   
2
2
2
b

x 
2

2
This is now a PTS!
So, it factors into this!
Ex 1:
In the following perfect square
trinomial, the constant term is missing.
x2 + 14x + ____
 Find the constant term by squaring half
the coefficient of the linear term.

 x2
 x2

+ 14x +
+ 14x +
𝑏 2
2
14 2
Factored form:
2
x2 + 14x + 49
PST
x  4
2
Example: Find the value of c that makes this
a PTS, then factor it.
𝑥2 − 3𝑥 + 𝑐
𝑥2
−3
− 3𝑥 +
2
𝑥2
PST:
2
9
− 3𝑥 +
4
2
Factored form:  x  3  or (2 x  3) 2

2
More examples:

Find a value for ‘c’ that will create a
perfect square trinomial, then factor it.
x2 + 20x + ___
 x2 - 4x + ___
 x2 + 5x + ___

100, (x+10)2
4, (x-2)2
25/4,
𝒙−
𝟓 𝟐 𝒐𝒓
𝟐
𝟐𝒙 − 𝟓
𝟐