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# 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.
(71 votes)
• Okay... he lost me at . 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)
(8 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)
• At , 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, at Sal 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)
• I have a question. When are you supposed to square root it and when are you not supposed to do that? For example: 9-x^2... you would square root that whole equation, right?
but for 4n^2-25... you wouldn't... is that because in the second expression, the variable came first?
(2 votes)
• You don't have equations. You have expressions. So, you would square root neither one.

Both your quadratics represent a difference of 2 squares and can be factored.
9-x^2 = (3-x)(3+x)
4n^2-25 = (2n-5)(2n+5)

Hope this helps.
(3 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.