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### Course: Integrated math 3>Unit 2

Lesson 5: Factoring using structure

# 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 :)
• 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.
• Okay... he lost me at . I have no idea what's going on, can someone help me?
• 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)
• How do you factor the difference of two squares?
• 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 ( - )
• How would i go about factoring 2r^2+3rs-2s^2
• 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

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)
• 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.
• 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.
• 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
• 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?
• 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?
• 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))
• 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
• 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?