Factoring Perfect Square Trinomials


What is a perfect square?If a number is squared the result is a perfect

square. Example 22=44 is a perfect square.Other examples:32=9 or 42=169 is a perfect square. 16 is a perfect square.


Here is a list of the perfect squares for the numbers 1-30.

12=1 112=121 212=441 22=4 122=144 222=484 32=9 132=169 232=529 42=16 142=196 242=576 52=25 152=225 252=625 62=36 162=256 262=676 72=49 172=289 272=729 82=64 182=324 282=784 92=81 192=361 292=841 102=100 202=400 302=900


When a variable is raised to an even power it is a perfect square.

Example: (x)(x)= x2

x2 is a perfect square.

(x3)(x3)= x6 or (x5)(x5)= x10

x6 and x10 are both perfect squares.


If a number or a variable is a perfect square the square root of the quantity is the number or variable that was squared to get the perfect square.

Example: Square 9.9x9 = 8181 is the perfect square.9 is the square root of 81.Example: Square x3

(x3) (x3) = x6 or (x3)2 = x6

x6 is the perfect square.x3 is the square root of x6


• Now we are ready to understand the term- perfect square trinomial.perfect square trinomial.

• The trinomial that results from squaring a binomial is a perfect square trinomialperfect square trinomial.

• Example: (x+7)2 = x2+14x+49• x2+14x+49 is a perfect square trinomial.• We know that a perfect square trinomial always results when a binomial is squared.

• The reverse is also true. • When we factor a perfect square trinomialperfect square trinomial

the result is always a squared binomial.


• Here are few examples:• Factor: x2+10x+25• Result: (x+5)2

• Check by multiplying• x2+10x+25• Factor: x2+2xy+y2

• (x+y) 2

• Check by multiplying.• x2+2xy+y2


• Not all trinomials are perfect square trinomials. • How do we recognize that a trinomial is a perfect square

trinomial.• The first and last terms of the trinomial must be perfect

squares and must be positive.• Example: x2+10x+25• What about the middle term? +10x• Take the square root of the first term x2 and get x.• Take the square root of the last term +25 and get 5.• Multiply (5)(x) and double the result. 10x. That is your middle

term.• Two times the product of the square roots of the first and last

terms will give the middle term.


• Here are some examples of trinomials that are perfect square trinomials.• 4x2 -20x +25• 2x• 5• (2x- 5)2

• 9x2 - 48xy + 64y2

• 3x• 8y• (3x-8y)2

• 2x3 +20x2y+50xy2

• Factor out the GCF• 2x(x2+10xy+25y2)• 2x• x• 5y• 2x(x+5y)2


Here are some examples that are not perfect square trinomials.x2+10x-25The last term is not positive.x2+2xy+2y2

The 2 in the last term is not a perfect square.4x2-10xy+25y2

The square root of the first term is 2x. The square root of the last term 5y.2(2x)(5y)= 20xy20xy = 10xy4x2-16xy+8y2

There is a common factor of four.4(x2- 4xy + 2y2)The last term of the trinomial is not a perfect square because the 2 in the last term

is not perfect square.To get more help go to the tutorial Practice- Factoring Perfect Square Trinomials

Factoring Perfect Square Trinomials

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