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## Algebra (all content)

### Course: Algebra (all content)>Unit 10

Lesson 17: Factoring quadratics: Difference of squares

# Factoring quadratics: Difference of squares

Learn how to factor quadratics that have the "difference of squares" form. For example, write x²-16 as (x+4)(x-4).
Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication.
In this article, we'll learn how to use the difference of squares pattern to factor certain polynomials. If you don't know the difference of squares pattern, please check out our video before proceeding.

## Intro: Difference of squares pattern

Every polynomial that is a difference of squares can be factored by applying the following formula:
start color #11accd, a, end color #11accd, squared, minus, start color #1fab54, b, end color #1fab54, squared, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis
Note that a and b in the pattern can be any algebraic expression. For example, for a, equals, x and b, equals, 2, we get the following:
\begin{aligned}\blueD{x}^2-\greenD{2}^2=(\blueD x+\greenD 2)(\blueD x-\greenD 2)\end{aligned}
The polynomial x, squared, minus, 4 is now expressed in factored form, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 2, right parenthesis. We can expand the right-hand side of this equation to justify the factorization:
\begin{aligned}(x+2)(x-2)&=x(x-2)+2(x-2)\\\\&=x^2-2x+2x-4\\ \\ &=x^2-4\end{aligned}
Now that we understand the pattern, let's use it to factor a few more polynomials.

## Example 1: Factoring $x^2-16$x, squared, minus, 16

Both x, squared and 16 are perfect squares, since x, squared, equals, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared and 16, equals, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared. In other words:
x, squared, minus, 16, equals, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared, minus, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared
Since the two squares are being subtracted, we can see that this polynomial represents a difference of squares. We can use the difference of squares pattern to factor this expression:
start color #11accd, a, end color #11accd, squared, minus, start color #1fab54, b, end color #1fab54, squared, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis
In our case, start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd and start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 4, end color #1fab54. Therefore, our polynomial factors as follows:
left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared, minus, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared, equals, left parenthesis, start color #11accd, x, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, right parenthesis, left parenthesis, start color #11accd, x, end color #11accd, minus, start color #1fab54, 4, end color #1fab54, right parenthesis
We can check our work by ensuring the product of these two factors is x, squared, minus, 16.

1) Factor x, squared, minus, 25.

2) Factor x, squared, minus, 100.

### Reflection question

3) Can we use the difference of squares pattern to factor x, squared, plus, 25?

## Example 2: Factoring $4x^2-9$4, x, squared, minus, 9

The leading coefficient does not have to equal to 1 in order to use the difference of squares pattern. In fact, the difference of squares pattern can be used here!
This is because 4, x, squared and 9 are perfect squares, since 4, x, squared, equals, left parenthesis, start color #11accd, 2, x, end color #11accd, right parenthesis, squared and 9, equals, left parenthesis, start color #1fab54, 3, end color #1fab54, right parenthesis, squared. We can use this information to factor the polynomial using the difference of squares pattern:
\begin{aligned}4x^2-9 &=(\blueD {2x})^2-(\greenD{3})^2\\ \\ &=(\blueD {2x}+\greenD 3)(\blueD {2x}-\greenD 3) \end{aligned}
A quick multiplication check verifies our answer.

4) Factor 25, x, squared, minus, 4.

5) Factor 64, x, squared, minus, 81.

6) Factor 36, x, squared, minus, 1.

## Challenge problems

7*) Factor x, start superscript, 4, end superscript, minus, 9.

8*) Factor 4, x, squared, minus, 49, y, squared.

## Want to join the conversation?

• If the question is x^2 + 25 its not factorable? Why?
• The difference of squares: (a+b)(a-b). x^2 + 25 is not factorable since you're adding 25, not subtracting. A positive multiplied by a negative is always a negative. If you were to factor it, you would have to use imaginary numbers such as i5. The factors of 25 are 5 and 5 besides 1 and itself. Since the formula: (a-b)(a+b), it uses a positive and negative sign, making the last term always a negative.
• How would you factor a "difference of squares" problem WITHOUT using this formula?

What fundamental steps or rules would you follow to do the expanding brackets steps in reverse?

= x^2 − 4​
= x^2 −2x +2x −4 #This step does not seem intuitive.
= x(x−2)+2(x−2)
= (x+2)(x−2)

Can you instead factor it by doing something with square roots?
• Your work above is correct. Maybe this will make your 1st step more intuitive. When you multiply 2 binomials, you usually get a trinomial (3 terms). With the difference of two squares, we get only a binomial (2 terms). Why does the middle term not exist? It's because it has a coefficient of 0. So, if you forget the pattern, stub in the middle term:
x^2 + 0x - 4
Then factor as usual. You need factors of -4 that add to 0, which leads you to the -2 and +2.

Hope this helps.
• i really don't understand factoring AT ALL even after watching the videos so can someone help me because i really suck at math! thank you in advance!
• Are you talking about just this video (difference of squares) or about any factoring at all? If it is factoring in general, you have to go through a sequence of factoring to support your learning. So just reaching out says that you are not as bad as you think. If you say all factoring, then we will start with a=1 and go one step at a time.
• Everything here is pretty easy, but why can't you factor x^2+25? what exactly prevents sums from working but differences work just fine? The included explanation doesn't explain in much detail
• Differences of perfect squares work because the middle term cancels out. (x+5)(x-5x^2-5x+5x-25=x^2-25. The pattern for squaring something like (x+5)^2=x^2+10x+25.
• What's the purpose of converting it to this form?
• It gives us the solutions to the quadratic directly! If I give you 4x^2 - 9 = 0, you can't really see the solutions. But, if I give you (2x-3)(2x+3) = 0 (which is the factored form), you can easily see that either 2x-3 = 0 or 2x+3 = 0. So, x can be 3/2 or -3/2. So, this form is important to figure the solutions to the quadratic, which would be important in the future lessons.

Another advantage to this form is in limits. Sometimes, a limit of a function can be indeterminate until you factor polynomials like this, which can cancel out terms and give us a finite limit.
• Does the order of the signs matter? I have gotten it right when I do (x+y)(x-y). If I did it (x-y)(x+y) would it be wrong?
• While it is the same answer, sometimes questions ask for a specific order.
• in the video how did he know what number to divide by ?
• I think he looked for their square roots. Like when he had 25 5*5 = 25 so that's how he figured it out. Unless that's not what you're talking about then I feel embarrassed