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.

How would you factor a "difference of squares" problem WITHOUT using this formula? What fundamental steps or rules would you follow to do the expanding brackets steps in reverse? = x^2 − 4 = x^2 −2x +2x −4 #This step does not seem intuitive. = x(x−2)+2(x−2) = (x+2)(x−2) Can you instead factor it by doing something with square roots?

Your work above is correct. Maybe this will make your 1st step more intuitive. When you multiply 2 binomials, you usually get a trinomial (3 terms). With the difference of two squares, we get only a binomial (2 terms). Why does the middle term not exist? It's because it has a coefficient of 0. So, if you forget the pattern, stub in the middle term: x^2 + 0x - 4 Then factor as usual. You need factors of -4 that add to 0, which leads you to the -2 and +2. Hope this helps.

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.

Everything here is pretty easy, but why can't you factor x^2+25? what exactly prevents sums from working but differences work just fine? The included explanation doesn't explain in much detail

Differences of perfect squares work because the middle term cancels out. (x+5)(x-5x^2-5x+5x-25=x^2-25. The pattern for squaring something like (x+5)^2=x^2+10x+25.

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

i cannot do this at all

Do you know your perfect squares? The basis of this video is that if you have a^2-b^2, it factors to (a+b)(a-b). This works because the middle terms, -ab+ab=0.

What's the purpose of converting it to this form?

It gives us the solutions to the quadratic directly! If I give you 4x^2 - 9 = 0, you can't really see the solutions. But, if I give you (2x-3)(2x+3) = 0 (which is the factored form), you can easily see that either 2x-3 = 0 or 2x+3 = 0. So, x can be 3/2 or -3/2. So, this form is important to figure the solutions to the quadratic, which would be important in the future lessons. Another advantage to this form is in limits. Sometimes, a limit of a function can be indeterminate until you factor polynomials like this, which can cancel out terms and give us a finite limit.

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.

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.

Main content

Algebra basics

Course: Algebra basics > Unit 7

Lesson 7: Factoring quadratics: Difference of squares

Factoring quadratics: Difference of squares

Google Classroom

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!]

Want to join the conversation?

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Elly Hamilton
Posted 4 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 + ...”
(6 votes)
Answer
- Anderson Le
  Posted 4 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...”
  (20 votes)
MathewJHill
Posted 4 months ago. Direct link to MathewJHill's post “How would you factor a "d...”
How would you factor a "difference of squares" problem WITHOUT using this formula?

What fundamental steps or rules would you follow to do the expanding brackets steps in reverse?

= x^2 − 4
= x^2 −2x +2x −4 #This step does not seem intuitive.
= x(x−2)+2(x−2)
= (x+2)(x−2)

Can you instead factor it by doing something with square roots?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Kim Seidel
  Posted 4 months ago. Direct link to Kim Seidel's post “Your work above is correc...”
  Your work above is correct. Maybe this will make your 1st step more intuitive. When you multiply 2 binomials, you usually get a trinomial (3 terms). With the difference of two squares, we get only a binomial (2 terms). Why does the middle term not exist? It's because it has a coefficient of 0. So, if you forget the pattern, stub in the middle term:
  x^2 + 0x - 4
  Then factor as usual. You need factors of -4 that add to 0, which leads you to the -2 and +2.
  
  Hope this helps.
  Button navigates to signup page
  (14 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
(6 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...”
  (12 votes)
Evie
Posted 3 months ago. Direct link to Evie's post “Everything here is pretty...”
Everything here is pretty easy, but why can't you factor x^2+25? what exactly prevents sums from working but differences work just fine? The included explanation doesn't explain in much detail
Button navigates to signup pageComment on Evie's post “Everything here is pretty...”
(4 votes)
Answer
- David Severin
  Posted 3 months ago. Direct link to David Severin's post “Differences of perfect sq...”
  Differences of perfect squares work because the middle term cancels out. (x+5)(x-5x^2-5x+5x-25=x^2-25. The pattern for squaring something like (x+5)^2=x^2+10x+25.
  Button navigates to signup page
  (9 votes)
rylan.wetsell
Posted 6 months ago. Direct link to rylan.wetsell's post “What's the purpose of con...”
What's the purpose of converting it to this form?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Venkata
  Posted 6 months ago. Direct link to Venkata's post “It gives us the solutions...”
  It gives us the solutions to the quadratic directly! If I give you 4x^2 - 9 = 0, you can't really see the solutions. But, if I give you (2x-3)(2x+3) = 0 (which is the factored form), you can easily see that either 2x-3 = 0 or 2x+3 = 0. So, x can be 3/2 or -3/2. So, this form is important to figure the solutions to the quadratic, which would be important in the future lessons.
  
  Another advantage to this form is in limits. Sometimes, a limit of a function can be indeterminate until you factor polynomials like this, which can cancel out terms and give us a finite limit.
  Comment on Venkata's post “It gives us the solutions...”
  (10 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...”
  (3 votes)
Fernando Garcia
Posted 6 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 ?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Melchi M. Katembo
  Posted 3 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
  (4 votes)
26tjpuffinberger
Posted 6 months ago. Direct link to 26tjpuffinberger's post “i cannot do this at all”
i cannot do this at all
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- David Severin
  Posted 6 months ago. Direct link to David Severin's post “Do you know your perfect ...”
  Do you know your perfect squares? The basis of this video is that if you have a^2-b^2, it factors to (a+b)(a-b). This works because the middle terms, -ab+ab=0.
  Button navigates to signup page
  (3 votes)
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
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- 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
  (3 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
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- 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)

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