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Algebra I (2018 edition)

Course: Algebra I (2018 edition)>Unit 15

Lesson 8: Factoring quadratics: Difference of squares

Difference of squares intro

When an expression can be viewed as the difference of two perfect squares, i.e. a²-b², then we can factor it as (a+b)(a-b). For example, x²-25 can be factored as (x+5)(x-5). This method is based on the pattern (a+b)(a-b)=a²-b², which can be verified by expanding the parentheses in (a+b)(a-b).

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• At , why is it called the Foil method? Does 'Foil' stand for something or is it just called that?
(38 votes)
• FOIL stands for "First, Outside, Inside, Last". You multiply together the first term in each binomial, the outside (leftmost and rightmost) terms, the inside terms, and the last term in each binomial. Take those four products, add them up, and you have the expanded expression.
(89 votes)
• What if we have a difference of squares like x^4 - y^4?
(15 votes)
• Great question! Always factor as much as possible. Whenever one of the resulting factors can be factored further, you must do so. For example:
``x⁴ - y⁴ = (x² + y²)(x² - y²) = (x² + y²)(x + y)(x - y)``
(43 votes)
• Well hello there friendly neighborhood precal student here
just wondering if someone can recommend me to a video or provide a little help here
the problem is x^2-10x-24
i had moved the 24 over from the right side
i have no idea what else to do afterward since all of these videos only explain it with an x^2 and a lone number

please help
thank you :)
(12 votes)
• The way to tell when you don't have a difference of 2 squares is if you can't find two perfect squares that are connected with a subtraction sign (the difference part of the name).

The following are valid examples of a difference of two squares, which can always be factored into a pair of conjugates (a ± b).
Notice that each term is a perfect square, and each starting expression is a mathematical difference:
`25x² - 49y²` factors into `(5x + 7y)(5x - 7y)`
`z⁴ - ¼` factors into `(z² + ½)(z² - ½)`
`9 - p²` factors into `(3 + p)(3 - p)`

As far as the expression that you were trying to factor `x² - 10x - 24 = (x + ?)(x - ?)` see below:
* The two ?s must be factors of -24. One's positive and one's negative so that their product is a negative number, the `-24`.
* Furthermore, they must sum to -10, the coefficient of the x term, so that the OUTER product and INNER product sum to the `-10x`.
* 2 and -12 satisfy these conditions, so `x² - 10x - 24 = (x + 2)(x - 12)`.
* Lastly, the factorization can be VERIFIED with binomial multiplication (FOIL - First Outer Inner Last):
``(x + 2)(x - 12) = x∙x - 12∙x + x∙2 - 12∙2 = x² - 12x + 2x - 24 = x² - 10x - 24 ``
(26 votes)
• One question... When you do have a difference of perfect squares problem...do you HAVE to use the a^2-b^2 method? Or can you still use the GCF method for them? It seems like every example I see, always uses the A^2-B^2 way to factor the difference of perfect squares...but the GCF way would work as well right?
(5 votes)
• The GCF method only helps if the two original terms share factors. Once you get the point where the terms are relatively prime, you must apply the difference of two squares method to obtain a fully factored answer. For example:
``36x² - 100y² = 4(9x²) + 4(-25y²) = find and factor out the GCF4(9x² - 25y²) = factor the difference of two squares4(3x + 5y)(3x - 5y)``
(11 votes)
• Is Difference of Squares is helpful when you try to find equations of conic sections?
(6 votes)
• when people write this symbol ^ what is it? Is it multiplication or divions
(1 vote)
• The symbol ^ generally means 'to the power of'. Multiplication is generally * and division by a /.

For example:

2^3 is 2 cubed, so 2 x 2 x 2 = 8
2*2 is 2 x 2 = 4
2/2 is 2 ÷ 2 = 1
(6 votes)
• 0:5 till maybe. Does a difference of squares have to be subtraction or can it be addition, because judging of the name "difference". So I'm thinking subtraction only, but I just want it clarify whether it can also be addition
(3 votes)
• Difference means subtraction, so it can only be a subtraction. If it is an addition such as x^2 + 1, think what that would do to the discriminant, b^2 - 4ac = 0^2 - 4(1)(1) = -4 which would mean you would have to take the root of a negative number, not allowed in the real domain. With c being a negative perfect square, the determinant is 0^2 - 4(a)(-c) so two negatives cancel, and since each of 4 and a and c are perfect squares, the root would be a whole number.
(2 votes)
• So im not sure really what im asking... but here I go.
What is the end goal of learning how to factor different equations? Why is learning so many ways to factor taking an entire section?

(I'm asking because being given a tool with no where to plug it into. It makes it harder for me to grasp the idea without seeing the big picture.)
(1 vote)
• Right now you are learning how to factor polynomial expressions. Later lessons will show you how to solve polynomial equations using factoring.

Other later lessons show you how to work with rational expressions and rational equations. These are expressions / equations that include fractions where the numerator and denominator are polynomials. All the operations we do with any type of fractions require the use of factors. We reduce fractions by removing common factors. We find LCDs to add/subtract fractions by using factors. We multiply and divide fractions using factors. So, to simplify, add, subtract, multiply and divide rational expressions, you need to know how to find factors of the polynomials in the numerators and denominators. We also use factors to help use solve rational equations.

Hope this helps.
(5 votes)
• Does the difference of squares method still work if it was (U^2)+(V^2) instead of (U^2)-(V^2)? If it doesn't work then why not?
(2 votes)
• The difference of squares factoring only works for (u^2)-(v^2), that is why it is called the difference of squares. (There is a way to factor (u^2)+(v^2), but it requires complex numbers. (u^2)+(v^2)=(u+v*i)*(u-v*i))
(3 votes)
• Why did you use the distributive property twice?
(2 votes)
• You use the distributive property to get the
answer for both x and -a
(2 votes)

Video transcript

- [Instructor] We're now going to explore factoring a type of expression called a difference of squares and the reason why it's called a difference of squares is 'cause it's expressions like x squared minus nine. This is a difference. We're subtracting between two quantities that are each squares. This is literally x squared. Let me do that in a different color. This is x squared minus three squared. It's the difference between two quantities that have been squared and it turns out that this is pretty straightforward to factor. And to see how it can be factored, let me pause there for a second and get a little bit of review of multiplying binomials. So put this on the back burner a little bit. Before I give you the answer of how you factor this, let's do a little bit of an exercise. Let's multiply x plus a times x minus a where a is some number. Now, we can use that, do that using either the FOIL method but I like just thinking of this as a distributive property twice. We could take x plus a and distribute it onto the x and onto the a. So when we multiply it by x, we would get x times x is x squared, a times x is plus ax and then when we multiply it by the negative a, well, it'll become negative a times x minus a squared. So these middle two terms cancel out and you are left with x squared minus a squared. You're left with a difference of squares. x squared minus a squared. So we have an interesting result right over here that x squared minus a squared is equal to, is equal to x plus a, x plus a times x minus a. And so we can use, and this is for any a. So we could use this pattern now to factor this. Here, what is our a? Our a is three. This is x squared minus three squared or we could say minus our a squared if we say three is a and so to factor it, this is just going to be equal to x plus our a which is three times x minus our a which is three. So x plus three times x minus three. Now, let's do some examples to really reinforce this idea of factoring differences of squares. So let's say we want to factor, let me say y squared minus 25 and it has to be a difference of squares. It doesn't work with a sum of squares. Well, in this case, this is going to be y and you have to confirm, okay, yeah, 25 is five squared and y squared is well, y squared. So this gonna be y plus something times y minus something and what is that something? Well, this right here is five squared so it's y plus five times y minus five and the variable doesn't have to come first. We could write 121 minus, I'll introduce a new variable, minus b squared. Well, this is a difference of squares because 121 is 11 squared. So this is going to be 11 plus something times 11 minus something and in this case, that something is going to be the thing that was squared. So 11 plus b times 11 minus b. So in general, if you see a difference of squares, one square being subtracted from another square and it could be a numeric perfect square or it could be a variable that has been squared that can be, that you could take the square root of. Well, then you could say, alright, well, that's just gonna be the first thing that squared plus the second thing that has been squared times the first thing that was squared minus the second thing that was squared. Now, some common mistakes that I've seen people do including my son when they first learned this is they say, okay, it's easy to recognize the difference of squares but then they say, oh, is this y squared plus 25 times y squared minus 25? No, the important thing to realize is is that what is getting squared? Over here, y is the thing getting squared and over here it is five that is getting squared. Those are the things that are getting squared in this difference of squares and so it's gonna be y plus five times y minus five. I encourage you to just try this out. We have a whole practice section on Khan Academy where you can do many many more of these to become familiar.