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### Course: Get ready for AP® Calculus>Unit 1

Lesson 8: Factoring quadratics with 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).

## Want to join the conversation?

• At , why is it called the Foil method? Does 'Foil' stand for something or is it just called that?
• 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.
• What if we have a difference of squares like x^4 - y^4?
• 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)``
• 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

thank you :)
• 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 ``
• 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?
• 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)``
• Is Difference of Squares is helpful when you try to find equations of conic sections?
• What’s a foil like I’m confused
• 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.
• 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
• 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.
• 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