Main content

## Get ready for Precalculus

### Course: Get ready for Precalculus > Unit 2

Lesson 5: Factoring using structure- Identifying quadratic patterns
- Identify quadratic patterns
- Factorization with substitution
- Factorization with substitution
- Factoring using the perfect square pattern
- Factoring using the difference of squares pattern
- Factor polynomials using structure

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Factoring using the difference of squares pattern

If we expand (a+b)(a-b) we will get a²-b². Factorization goes the other way: suppose we have an expression that is the difference of two squares, like x²-25 or 49x²-y², then we can factor is using the roots of those squares. For example, x²-25 can be factored as (x+5)(x-5). This is an extremely useful method that is used throughout math. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Given 20* 20 = 400, use the difference of two squares to determine 19*21.

im stuck on this question, my thoughts are its a trick question because 19 and 21 are not perfect squares but is somone could help me that would be great :)(25 votes)- The difference of squares magic, math trick, or math principle, actually works even better than just when the numbers are only one away from the known square.

20*20 = 400

19*21 = 400-(1*1) or 20^2-1^2 = 399

18*22 = 400-(2*2) or 20^2-2^2 = 396

17*23 = 400-(3*3) or 20^2 - 3^2 = 391

To find y * z, if integer x is half way between y and z you can square x and then subtract the square of (z-x)

The "difference of two squares" can help sometimes.(69 votes)

- Okay... he lost me at0:14. I have no idea what's going on, can someone help me?(9 votes)
- a difference of square is a binomial in which both the terms are perfect squares and they are subtracted

a2-b2

if you have a difference of squares expression here is how you would factor it

a2-b2=(a+b)(a-b)

in this case it is

x2-49y2

a=x

b=7y

x2-49y2=(x+7y)(x-7y)(11 votes)

- How do you factor the difference of two squares?(6 votes)
- In the form x^2 - y^2, it is (x+y)(x-y).

Example: 4x^2 - 25

divide the coefficient of x by 2

take the square root of 25

One expression is ( + ) the other is ( - )

answer: (2x+5)(2x-5)(7 votes)

- How would i go about factoring 2r^2+3rs-2s^2(4 votes)
- 2r^2 + 3rs - 2s^2 =

2(r^2 + 3/2 rs - s^2) =

2(r + 2s)(r - 1/2 s) =

(r + 2s)(2r - s)

----------------------------

Attempting to explain the second step :

2(r^2 + 3/2 rs - s^2) =

2(ar + bs)(cr + ds)

ac = 1

bd = -1

ad + bc = 3/2

Trying with a=1:

ac = 1 so c=1

so, since a=1 and c=1

b + d = 3/2

we already knew that bd = -1, so what numbers add up to 3/2 and multiplies to -1 ?

trying with b=1: d=-1 and d=1/2. No go.

trying with b=2: d=-1/2 and d=-1/2. That works.

Now we have a=1, c=1, b=2, d=-1/2, so

2(ar + bs)(cr + ds) =

2(r + 2s)(r - 1/2 s) =

(r + 2s)(2r - s)(7 votes)

- I can't find any way to factor the following problem:

81p^2 − 144pq + 64q^2

The is no number that goes into both of them and there are two variables

please help me.(3 votes)- Your trinomial is a perfect square. It factors into:

(9p-8q)(9p-8q)

You can learn about perfect square trinomials at https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:factor-perfect-squares/v/perfect-square-factorization-intro

Hope this helps.(4 votes)

- At1:06, wouldn't it be "x minus 7y, the whole thing squared" instead of "x squared minus 7y, the whole thing squared"? That's equal to x to the fourth minus 7y squared.(5 votes)
- You are right, at1:06Sal might have meant to say "x minus 7y, the whole thing squared," which would have been (x - 7y), and when this binomial is squared, it results in x^2 - 7y^2.(2 votes)

- In the beginning of this video the narrator introduces us to an equation that is x squared-49y squared. Then he equates it to x squared-7y squared. How did he get that?(2 votes)
- He found the square root of the 49y^2! You figure out what number squared would make 49y, which is 7y! :)(1 vote)

- I understand the concept, but I'm confused how I'm supposed to apply what Sal is saying in the video to my problem. Here it is=

The rectangle below (I know, I cant copy and paste pictures) has an area of x^2 - 144 square meters and a length of x+2, What expression represents the width of the rectangle?(3 votes)- You know that area of rectangle = length × Width, Right ?

And you have the area and the length so (x+2)Width = x^2 - 144

Width = (x^2 - 144)/(x+2)

by simplifying, the answer will be: Width = x-2-(140/(x+2))(2 votes)

- What's the difference between a perfect square and not a perfect square?(1 vote)
- A perfect square is a number such as 9 or 25. It means that when you take the square root of the number, it simplifies out to a whole number without decimals. Basically, a perfect square is like the product of 2 times 2 or 3 times 3 or 4 times 4. A non-perfect square comes out to be any decimal or an irrational number like square root of 7 or 13(4 votes)

- Is a^2-b^2 the same as (a-b)^2. If not could you please explain why? This doubt has been troubling me for a few days. Thanks(1 vote)
- If you try this with numbers it is easy to see why it does not work. It is sort of related to order of operations. Let's say a is 5 and b is 2.

a² – b² = 5² – 2² = 25 – 4 = 21

Now, if you subtract first...

(a – b)² = (5 – 2)² = (3)² = 9

You can see that since you subtracted first, you end up squaring a much smaller number, so the values are very different. Just like changing the order of operations in most problems will change the value.(4 votes)

## Video transcript

Factor x squared
minus 49y squared. So what's interesting
here is that well x squared is clearly
a perfect square. It's the square of x. And 49y squared is
also a perfect square. It's the square of 7y. So it looks like we might
have a special form here. And to remind
ourselves, let's think about what happens if we take
a plus b times a minus b. I'm just doing it
in the general case so we can see a pattern here. So over here, this
would be a times a, which would be a squared
plus a times negative b, which would be negative ab plus
b times a or a times b again, which would be ab. And then you have
b times negative b, so it would b minus b squared. Now these middle two
terms cancel out. Negative ab plus
ab, they cancel out and you're left with just
a squared minus b squared. And that's the exact
pattern we have here. We have an a squared
minus a b squared. So in this case, a is equal
to x and b is equal to 7y. So we have x squared minus
7y, the whole thing squared. So we can expand this as
the difference of squares, or actually this
thing right over here is the difference of squares. So we expand this like this. So this will be equal to x
plus 7y times x minus 7y. And once again,
we're just pattern matching based on this
realization right here. If I take a plus
b times a minus b, I get a difference of squares. This is a difference of squares. So when I factor
it, it must come out to the result of
something that looks like a plus b times a minus b
or x plus 7y times x minus 7y.