If the question is x^2 + 25 its not factorable? Why?

The difference of squares: (a+b)(a-b). x^2 + 25 is not factorable since you're adding 25, not subtracting. A positive multiplied by a negative is always a negative. If you were to factor it, you would have to use imaginary numbers such as i5. The factors of 25 are 5 and 5 besides 1 and itself. Since the formula: (a-b)(a+b), it uses a positive and negative sign, making the last term always a negative.

i really don't understand factoring AT ALL even after watching the videos so can someone help me because i really suck at math! thank you in advance!

Are you talking about just this video (difference of squares) or about any factoring at all? If it is factoring in general, you have to go through a sequence of factoring to support your learning. So just reaching out says that you are not as bad as you think. If you say all factoring, then we will start with a=1 and go one step at a time.

Does the order of the signs matter? I have gotten it right when I do (x+y)(x-y). If I did it (x-y)(x+y) would it be wrong?

While it is the same answer, sometimes questions ask for a specific order.

in the video how did he know what number to divide by ?

I think he looked for their square roots. Like when he had 25 5*5 = 25 so that's how he figured it out. Unless that's not what you're talking about then I feel embarrassed

So when you have 25x-4 you have to make the 4 into a 2

Yes, because the square root of 4 is 2. You should also notice the 25x^2. The square root of 25x^2 is 5x.

factoris 27-3x to the power of 2

Same as your other question. Factor out the common factor as your first step. Then you will have a difference of 2 squares. Give it a try.

How would I solve for a question like xp^2-4x and give a number for x to make the binomial a difference of perfect squares?

You aren't solving, you are factoring. The first step you should always try when factoring is to look for a common factor. Your binomials has a common factor of X. Factor it out using the distributive property. x (p^2 - 4) You now have a binomial in the parentheses that is a difference of 2 squares. When you factor it, your factors become: x(p-2)(p+2) Hope this helps.

Main content

Algebra basics

Course: Algebra basics > Unit 7

Lesson 7: Factoring quadratics: Difference of squares

Factoring quadratics: Difference of squares

CCSS.Math:

Learn how to factor quadratics that have the "difference of squares" form. For example, write x²-16 as (x+4)(x-4).

Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication.

In this article, we'll learn how to use the difference of squares pattern to factor certain polynomials. If you don't know the difference of squares pattern, please check out our video before proceeding.

Intro: Difference of squares pattern

Every polynomial that is a difference of squares can be factored by applying the following formula:

start color #11accd, a, end color #11accd, squared, minus, start color #1fab54, b, end color #1fab54, squared, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis

Note that a and b in the pattern can be any algebraic expression. For example, for a, equals, x and b, equals, 2, we get the following:

\begin{aligned}\blueD{x}^2-\greenD{2}^2=(\blueD x+\greenD 2)(\blueD x-\greenD 2)\end{aligned}

The polynomial x, squared, minus, 4 is now expressed in factored form, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 2, right parenthesis. We can expand the right-hand side of this equation to justify the factorization:

\begin{aligned}(x+2)(x-2)&=x(x-2)+2(x-2)\\\\&=x^2-2x+2x-4\\ \\ &=x^2-4\end{aligned}

Now that we understand the pattern, let's use it to factor a few more polynomials.

Example 1: Factoring x, squared, minus, 16

Both x, squared and 16 are perfect squares, since x, squared, equals, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared and 16, equals, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared. In other words:

x, squared, minus, 16, equals, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared, minus, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared

Since the two squares are being subtracted, we can see that this polynomial represents a difference of squares. We can use the difference of squares pattern to factor this expression:

In our case, start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd and start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 4, end color #1fab54. Therefore, our polynomial factors as follows:

left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared, minus, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared, equals, left parenthesis, start color #11accd, x, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, right parenthesis, left parenthesis, start color #11accd, x, end color #11accd, minus, start color #1fab54, 4, end color #1fab54, right parenthesis

We can check our work by ensuring the product of these two factors is x, squared, minus, 16.

[I'd like to see this, please!]

Check your understanding

1) Factor x, squared, minus, 25.

(Choice A)
left parenthesis, x, minus, 5, right parenthesis, left parenthesis, x, minus, 5, right parenthesis
(Choice B)
left parenthesis, x, plus, 5, right parenthesis, left parenthesis, x, plus, 5, right parenthesis
(Choice C)
left parenthesis, x, plus, 5, right parenthesis, left parenthesis, x, minus, 5, right parenthesis

[I need help!]

2) Factor x, squared, minus, 100.

[I need help!]

Reflection question

3) Can we use the difference of squares pattern to factor x, squared, plus, 25?

[I need help!]

Example 2: Factoring 4, x, squared, minus, 9

The leading coefficient does not have to equal to 1 in order to use the difference of squares pattern. In fact, the difference of squares pattern can be used here!

This is because 4, x, squared and 9 are perfect squares, since 4, x, squared, equals, left parenthesis, start color #11accd, 2, x, end color #11accd, right parenthesis, squared and 9, equals, left parenthesis, start color #1fab54, 3, end color #1fab54, right parenthesis, squared. We can use this information to factor the polynomial using the difference of squares pattern:

\begin{aligned}4x^2-9 &=(\blueD {2x})^2-(\greenD{3})^2\\ \\ &=(\blueD {2x}+\greenD 3)(\blueD {2x}-\greenD 3) \end{aligned}

A quick multiplication check verifies our answer.

[I'd like to see this, please!]

Check your understanding

4) Factor 25, x, squared, minus, 4.

(Choice A)
left parenthesis, 25, x, minus, 2, right parenthesis, left parenthesis, x, plus, 2, right parenthesis
(Choice B)
left parenthesis, 25, x, plus, 2, right parenthesis, left parenthesis, x, minus, 2, right parenthesis
(Choice C)
left parenthesis, 5, x, plus, 2, right parenthesis, left parenthesis, 5, x, minus, 2, right parenthesis
(Choice D)
left parenthesis, 25, x, plus, 4, right parenthesis, left parenthesis, 25, x, minus, 4, right parenthesis

[I need help!]

5) Factor 64, x, squared, minus, 81.

[I need help!]

6) Factor 36, x, squared, minus, 1.

[I need help!]

Challenge problems

7*) Factor x, start superscript, 4, end superscript, minus, 9.

[I need help!]

8*) Factor 4, x, squared, minus, 49, y, squared.

[I need help!]

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Elly Hamilton
Posted 3 years ago. Direct link to Elly Hamilton's post “If the question is x^2 + ...”
If the question is x^2 + 25 its not factorable? Why?
Button navigates to signup pageComment on Elly Hamilton's post “If the question is x^2 + ...”
(5 votes)
Answer
- Anderson Le
  Posted 3 years ago. Direct link to Anderson Le's post “The difference of squares...”
  The difference of squares: (a+b)(a-b). x^2 + 25 is not factorable since you're adding 25, not subtracting. A positive multiplied by a negative is always a negative. If you were to factor it, you would have to use imaginary numbers such as i5. The factors of 25 are 5 and 5 besides 1 and itself. Since the formula: (a-b)(a+b), it uses a positive and negative sign, making the last term always a negative.
  Comment on Anderson Le's post “The difference of squares...”
  (13 votes)
Brittne Wiley
Posted 3 years ago. Direct link to Brittne Wiley's post “i really don't understand...”
i really don't understand factoring AT ALL even after watching the videos so can someone help me because i really suck at math! thank you in advance!
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- David Severin
  Posted 3 years ago. Direct link to David Severin's post “Are you talking about jus...”
  Are you talking about just this video (difference of squares) or about any factoring at all? If it is factoring in general, you have to go through a sequence of factoring to support your learning. So just reaching out says that you are not as bad as you think. If you say all factoring, then we will start with a=1 and go one step at a time.
  Comment on David Severin's post “Are you talking about jus...”
  (11 votes)
purpleabb
Posted 5 years ago. Direct link to purpleabb's post “Does the order of the sig...”
Does the order of the signs matter? I have gotten it right when I do (x+y)(x-y). If I did it (x-y)(x+y) would it be wrong?
Button navigates to signup pageComment on purpleabb's post “Does the order of the sig...”
(5 votes)
Answer
- David Severin
  Posted 5 years ago. Direct link to David Severin's post “While it is the same answ...”
  While it is the same answer, sometimes questions ask for a specific order.
  Comment on David Severin's post “While it is the same answ...”
  (2 votes)
Fernando Garcia
Posted 5 years ago. Direct link to Fernando Garcia's post “in the video how did he k...”
in the video how did he know what number to divide by ?
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(4 votes)
Answer
- Melchi M. Katembo
  Posted 2 years ago. Direct link to Melchi M. Katembo's post “I think he looked for the...”
  I think he looked for their square roots. Like when he had 25 5*5 = 25 so that's how he figured it out. Unless that's not what you're talking about then I feel embarrassed
  Button navigates to signup page
  (2 votes)
shackelforddo2023
Posted 3 years ago. Direct link to shackelforddo2023's post “So when you have 25x-4 yo...”
So when you have 25x-4 you have to make the 4 into a 2
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(2 votes)
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- Anthony Silva
  Posted 3 years ago. Direct link to Anthony Silva's post “Yes, because the square r...”
  Yes, because the square root of 4 is 2. You should also notice the 25x^2. The square root of 25x^2 is 5x.
  Button navigates to signup page
  (2 votes)
benjamin.kolkert
Posted 3 years ago. Direct link to benjamin.kolkert's post “how do you explain differ...”
how do you explain difference of squares to someone in words
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matsedisoraseipone765
Posted 2 years ago. Direct link to matsedisoraseipone765's post “factoris 27-3x to the pow...”
factoris 27-3x to the power of 2
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(1 vote)
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- Kim Seidel
  Posted 2 years ago. Direct link to Kim Seidel's post “Same as your other questi...”
  Same as your other question. Factor out the common factor as your first step. Then you will have a difference of 2 squares. Give it a try.
  Button navigates to signup page
  (2 votes)
tcohen21
Posted 4 years ago. Direct link to tcohen21's post “Why can’t you factorise t...”
Why can’t you factorise the difference of squares in the expression: (4a^2-9c^2)/ 12abc?
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(1 vote)
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MJ
Posted 2 years ago. Direct link to MJ's post “How would I solve for a q...”
How would I solve for a question like xp^2-4x and give a number for x to make the binomial a difference of perfect squares?
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(0 votes)
Answer
- Kim Seidel
  Posted 2 years ago. Direct link to Kim Seidel's post “You aren't solving, you a...”
  You aren't solving, you are factoring.
  The first step you should always try when factoring is to look for a common factor. Your binomials has a common factor of X. Factor it out using the distributive property.
  x (p^2 - 4)
  You now have a binomial in the parentheses that is a difference of 2 squares. When you factor it, your factors become:
  x(p-2)(p+2)
  Hope this helps.
  Button navigates to signup page
  (3 votes)
Rabi
Posted 2 years ago. Direct link to Rabi's post “Is it possible to create ...”
Is it possible to create a formula like this:
(a^2-b^2)=(√a+√b)(√a-√b)
and if so what will be the domain of it.
Thanks
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Algebra basics

Course: Algebra basics > Unit 7

Intro: Difference of squares pattern

Example 1: Factoring x2−16x^2-16x2−16x, squared, minus, 16

Check your understanding

Reflection question

Example 2: Factoring 4x2−94x^2-94x2−94, x, squared, minus, 9

Check your understanding

Challenge problems

Want to join the conversation?

Example 1: Factoring x, squared, minus, 16

Example 2: Factoring 4, x, squared, minus, 9