In number 5, where did the -4x+1x come from? -3x doesn't split up into -4x+1x and multiply out to -2. Help!

Hi, ShadowFax5! Be careful there: it does not have to multiply out to -2. You've missed a step. It has to multiply out to -4, since the leading term is 2 (different, thus, than 1) and it has to be multiplied by the last term -2, remember? Take a look at how I solved it: `8x^2-12x-8` Factor out 4: `4(2x-3x-2)` Leading term `2` times last term `-2` = `-4` and middle term `-3`. Find: `A+B = -3` that is `-4+1` `A*B = -4` that is `-4*1` Group it: `4[(2x^2-4x)+(1x-2)]` Factor out the CGF of the groups: `4[2x(x-2)+1(x-2)]` Regroup it by factoring out the (x-2): `4(2x+1)(x-2)` There you have it. Hope it helps and if someone find any error, please comment! Cheers!

i don't understand how the answer to the last question (#7) is 3(x^2 + 9) isn't x^2 + 9 a sum of squares so shouldn't it be 3(x+3)^2 ?

3(x+3)^2 isn't equivalent to x^2+9. The x^2+9 does look like a sum of squares at first, but to be that it would need to factor to (x+3)(x+3); (x+3)(x+3) FOILed is x^2+6x+9, not x^2+9.

can there be no way to factor a problem?

Yes some problems are un-factorable in the real domain. If you look at b^2 - 4ac, if this is positive you have 2 factors, if it is 0 you have one factor, and if it is negative, is does not have factors in real domain. The first crosses x axis at two places, the second has the vertex on the x-axis, and the third has the vertex above if open upward or below if open downward and never crosses the x axis.

Is a 4th dimensional graph a thing?

It is, but very hard too graph and not encountered that much (if at all) in real life.

Just to confirm Is (x-4)^2 = (4-x)^2

Yes, you are correct here is why: (x-4)^2 = x^2 + 2(x)(-4) + 16 = x^2 - 8x + 16 (4-x)^2 = x^2 + 2(-x)(4) + 16 = x^2 - 8x + 16 Notice, however, that in the second expression it is -x and 4 while in the first it is x and -4.

I noticed that the formula _x^2 + (a+b)x + ab_ doesn't make an appearance here from the Factoring simple quadratics review article but formula _x^2 + 2abx + b^2 does_. I'm studying for the unit test in the Algebra basics course but cannot figure out which formula applies when factoring. Does it have to do with when the x terms and constant term aren't perfect squares or when the x^2 term is not equal to 1? Thanks in advance.

`x^2 (a+b)x + ab` is the general rule for factorising, basically signifying that you find 2 numbers a and b which add to the coefficient of x and multiply to the constant term, to factorise to `(x + a)(x + b) = 0`. `x^2 + 2bx + b^2` (not `x^2 + 2abx + b^2`) is the expansion of (x + b)^2, a perfect square; this is a certain situation in factorising where it's a perfect square. It is useful to know for factoring but is most important in the _completing the square_ method.

So, question 6 says to factor 56-18x+x^2 completely. After placing in standard form: x^2 -18X +56, the next step should be to find the GCF. 2( x^2 -9x + 28). After this, there is no a+b=-9 and a * b = 28. Factors of 28 are 1,28, 2, 14, 4, 7. So none of them work. However, apparently, when I asked for the help, they just skipped right past the GCF that they had taught us to try first and went right on to the sum/product step. Does anyone else feel confused about this?

The polynomial has no common factor other than 1. In order for there to have been a common factor of 2, the problem would have been: 2x^2-18x+56. Yes, you should always look for a GCF. But *all terms* need to be evenly divisible by the value you pick. x^2 does not divide evenly by 2 in your problem, so the GCF=1 and there is no need to factor out a GCF. Hope this help.s

Main content

Course: Integrated math 2 > Unit 2

Lesson 9: Strategy in factoring quadratics

Factoring quadratics in any form

Google Classroom

Tie together everything you learned about quadratic factorization in order to factor various quadratic expressions of any form.

What you need to know for this lesson

The following factoring methods will be used in this lesson:

What you will learn in this lesson

In this article, you will practice putting these methods together to completely factor quadratic expressions of any form.

Intro: Review of factorization methods

Method	Example	When is it applicable?
Factoring out common factors	$\begin{aligned} 6 x^{2} + 3 x \\ = 3 x (2 x + 1) \end{aligned}$ ‍	If each term in the polynomial shares a common factor.
The sum-product pattern	$\begin{aligned} x^{2} + 7 x + 12 \\ = (x + 3) (x + 4) \end{aligned}$ ‍	If the polynomial is of the form $x^{2} + b x + c$ ‍ and there are factors of $c$ ‍ that add up to $b$ ‍.
The grouping method	$\begin{aligned} 2 x^{2} + 7 x + 3 \\ = 2 x^{2} + 6 x + 1 x + 3 \\ = 2 x (x + 3) + 1 (x + 3) \\ = (x + 3) (2 x + 1) \end{aligned}$ ‍	If the polynomial is of the form $a x^{2} + b x + c$ ‍ and there are factors of $a c$ ‍ that add up to $b$ ‍.
Perfect square trinomials	$\begin{aligned} x^{2} + 10 x + 25 \\ = (x + 5)^{2} \end{aligned}$ ‍	If the first and last terms are perfect squares and the middle term is twice the product of their square roots.
Difference of squares	$\begin{aligned} x^{2} - 9 \\ = (x - 3) (x + 3) \end{aligned}$ ‍	If the expression represents a difference of squares.

Putting it all together

In practice, you'll rarely be told what type of factoring method(s) to use when encountering a problem. So it's important that you develop some sort of checklist to use to help make the factoring process easier.

Here's one example of such a checklist, in which a series of questions is asked in order to determine how to factor the quadratic polynomial.

Factoring quadratic expressions

Before starting any factoring problem, it is helpful to write your expression in standard form.

Once this is the case, you can proceed to the following list of questions:

Question 1: Is there a common factor?
If no, move onto Question 2. If yes, factor out the GCF and continue to Question 2.

Factoring out the GCF is a very important step in the factoring process, as it makes the numbers smaller. This, in turn, makes it easier to recognize patterns!

Question 2: Is there a difference of squares (i.e. $x^{2} - 16$ ‍ or $25 x^{2} - 9$ ‍)?
If a difference of squares pattern occurs, factor using the pattern

a^{2} - b^{2} = (a + b) (a - b)

. If not, move on to Question 3.

Question 3: Is there a perfect square trinomial (i.e. $x^{2} - 10 x + 25$ ‍ or $4 x^{2} + 12 x + 9$ ‍)?
If a perfect square trinomial is present, factor using the pattern

a^{2} \pm 2 a b + b^{2} = (a \pm b)^{2}

. If not, move on to Question 4.

Question 4:

a.) Is there an expression of the form $x^{2} + b x + c$ ‍?
If no, move on to Question 5. If yes, move on to b).

b.) Are there factors of $c$ ‍ that sum to $b$ ‍?
If yes, then factor using the sum-product pattern. Otherwise, the quadratic expression cannot be factored further.

Question 5: Are there factors of $a c$ ‍ that add up to $b$ ‍?
If you've gotten this far, the quadratic expression must be of the form

a x^{2} + b x + c

where

a \neq 1

. If there are factors of

a c

that add up to

b

, factor using the grouping method. If not, the quadratic expression cannot be factored further.

Following this checklist will help to ensure that you've factored the quadratic completely!

When you factor completely, you continue factoring until all factors cannot be factored further.

For example, consider the polynomial

2 x^{2} - 4 x - 16

. This can be factored by taking out a common factor of

2

$2 x^{2} - 4 x - 16 = 2 (x^{2} - 2 x - 8)$ ‍

However, this is not completely factored since

x^{2} - 2 x - 8

can be factored further. Adding in this last step, the process looks like:

$\begin{aligned} 2 x^{2} - 4 x - 16 & = 2 (x^{2} - 2 x - 8) \\ = 2 (x - 4) (x + 2) \end{aligned}$ ‍

At this point, we cannot factor further since we have expressed the polynomial as a product of linear factors with no GCFs.

So while

2 (x^{2} - 2 x - 8)

is a factorization of

2 x^{2} - 4 x - 16

, the complete factorization is

2 (x - 4) (x + 2)

With this in mind, let's try a few examples.

Example 1: Factoring $5 x^{2} - 80$ ‍

Notice that the expression is already in standard form. We can proceed to the checklist.

Question 1: Is there a common factor?
Yes. The GCF of

5 x^{2}

and

80

5

. We can factor this out as follows:

$5 x^{2} - 80 = 5 (x^{2} - 16)$ ‍

Question 2: Is there a difference of squares?
Yes.

x^{2} - 16 = (x)^{2} - (4)^{2}

. We can use the difference of squares pattern to continue factoring the polynomial as shown below.

$\begin{aligned} = 5 ((x)^{2} - (4)^{2}) \\ = 5 (x + 4) (x - 4) \end{aligned}$ ‍

There are no more quadratics in the expression. We have completely factored the polynomial.

In conclusion,

5 x^{2} - 80 = 5 (x + 4) (x - 4)

Example 2: Factoring $4 x^{2} + 12 x + 9$ ‍

The quadratic expression is again in standard form. Let's start the checklist!

Question 1: Is there a common factor?
No. The terms

4 x^{2}

12 x

and

9

do not share a common factor. Next question.

Question 2: Is there a difference of squares?
No. There’s an

x

-term so this cannot be a difference of squares. Next question.

Question 3: Is there a perfect square trinomial?
Yes. The first term is a perfect square since

4 x^{2} = (2 x)^{2}

, and the last term is a perfect square since

9 = (3)^{2}

. Also, the middle term is twice the product of the numbers that are squared since

12 x = 2 (2 x) (3)

We can use the perfect square trinomial pattern to factor the quadratic.

$\begin{aligned} 4 x^{2} + 12 x + 9 \\ = (2 x)^{2} + 2 (2 x) (3) + (3)^{2} \\ = (2 x + 3)^{2} \end{aligned}$ ‍

In conclusion,

4 x^{2} + 12 x + 9 = (2 x + 3)^{2}

Example 3: Factoring $12 x - 63 + 3 x^{2}$ ‍

This quadratic expression is not currently in standard form. We can rewrite it as

3 x^{2} + 12 x - 63

and then proceed through the checklist.

Question 1: Is there a common factor?
Yes. The GCF of

3 x^{2}

12 x

and

63

3

. We can factor this out as follows:

$3 x^{2} + 12 x - 63 = 3 (x^{2} + 4 x - 21)$ ‍

Question 2: Is there a difference of squares?
No. Next question.

Question 3: Is there a perfect square trinomial?
No. Notice that

21

is not a perfect square, so this cannot be a perfect square trinomial. Next question.

Question 4a: Is there an expression of the form $x^{2} + b x + c$ ‍?
Yes. The resulting quadratic,

x^{2} + 4 x - 21

, is of this form.

Question 4b: Are there factors of $c$ ‍ that add up to $b$ ‍?
Yes. Specifically, there are factors of

- 21

that add up to

4

Since

7 \cdot (- 3) = - 21

and

7 + (- 3) = 4

, we can continue to factor as follows:

$\begin{aligned} = 3 (x^{2} + 4 x - 21) \\ = 3 (x + 7) (x - 3) \end{aligned}$ ‍

In conclusion,

3 x^{2} + 12 x - 63 = 3 (x + 7) (x - 3)

Example 4: Factoring $4 x^{2} + 18 x - 10$ ‍

Notice that this quadratic expression is already in standard form.

Question 1: Is there a common factor?
Yes. The GCF of

4 x^{2}

18 x

and

10

2

. We can factor this out as follows:

$4 x^{2} + 18 x - 10 = 2 (2 x^{2} + 9 x - 5)$ ‍

Question 2: Is there a difference of squares?
No. Next question.

Question 3: Is there a perfect square trinomial?
No. Next question.

Question 4a: Is there an expression of the form $x^{2} + b x + c$ ‍?
No. The leading coefficient on the quadratic factor is

2

. Next question.

Question 5: Are there factors of $a c$ ‍ that add up to $b$ ‍?
The resulting quadratic expression is

2 x^{2} + 9 x - 5

, and so we want to find factors of

2 \cdot (- 5) = - 10

that add up to

9

Since

(- 1) \cdot 10 = - 10

and

(- 1) + 10 = 9

, the answer is yes.

We can now write the middle term as

- 1 x + 10 x

and use grouping to factor:

\begin{aligned} 2 (2 x^{2} + 9 x - 5) \\ = 2 (2 x^{2} - 1 x + 10 x - 5) & Split up middle term \\ = 2 ((2 x^{2} - 1 x) + (10 x - 5)) & Group terms \\ = 2 (x (2 x - 1) + 5 (2 x - 1)) & Factor out GCFs \\ = 2 (2 x - 1) (x + 5) & Factor out 2 x - 1 \end{aligned}

Check your understanding

1) Factor $2 x^{2} + 4 x - 16$ ‍ completely.

2) Factor $3 x^{2} - 60 x + 300$ ‍ completely.

3) Factor $72 x^{2} - 2$ ‍ completely.

4) Factor $5 x^{2} + 5 x + 15$ ‍ completely.

5) Factor $8 x^{2} - 12 x - 8$ ‍ completely.

6) Factor $56 - 18 x + x^{2}$ ‍ completely.

7) Factor $3 x^{2} + 27$ ‍ completely.

\begin{aligned} 3 x^{2} + 27 \\ = 3 (x^{2} + 9) & Factor out 3 \end{aligned}

The expression

x^{2} + 9

cannot be factored further.

We can make sure this is so by writing the expression as

x^{2} + 0 x + 9

. Since there are no two numbers whose product is

9

that add up to

0

, this expression is not factorable.

Another way to think about it is that

x^{2} + 9

is a sum of squares, and not a difference of squares. A sum of squares is never factorable over the real numbers.

Want to join the conversation?

Sort by:

ShadowFax5
Posted 8 years ago. Direct link to ShadowFax5's post “In number 5, where did th...”
In number 5, where did the -4x+1x come from? -3x doesn't split up into -4x+1x and multiply out to -2. Help!
Button navigates to signup pageComment on ShadowFax5's post “In number 5, where did th...”
(24 votes)
Answer
- Beorn MacRupert
  Posted 8 years ago. Direct link to Beorn MacRupert's post “Hi, ShadowFax5! Be caref...”
  Hi, ShadowFax5!
  
  Be careful there: it does not have to multiply out to -2. You've missed a step. It has to multiply out to -4, since the leading term is 2 (different, thus, than 1) and it has to be multiplied by the last term -2, remember? Take a look at how I solved it:
  8x^2-12x-8
  Factor out 4:
  4(2x-3x-2)
  Leading term 2 times last term -2 = -4 and middle term -3. Find:
  A+B = -3 that is -4+1
  A*B = -4 that is -4*1
  Group it:
  4[(2x^2-4x)+(1x-2)]
  Factor out the CGF of the groups:
  4[2x(x-2)+1(x-2)]
  Regroup it by factoring out the (x-2):
  4(2x+1)(x-2)
  
  There you have it.
  Hope it helps and if someone find any error, please comment!
  Cheers!
  Comment on Beorn MacRupert's post “Hi, ShadowFax5! Be caref...”
  (40 votes)
Anastasia.Theys
Posted 7 years ago. Direct link to Anastasia.Theys's post “i don't understand how th...”
i don't understand how the answer to the last question (#7) is 3(x^2 + 9)
isn't x^2 + 9 a sum of squares so shouldn't it be 3(x+3)^2 ?
Button navigates to signup pageButton navigates to signup page
(8 votes)
Answer
- Ellen Wight
  Posted 7 years ago. Direct link to Ellen Wight's post “3(x+3)^2 isn't equivalent...”
  3(x+3)^2 isn't equivalent to x^2+9. The x^2+9 does look like a sum of squares at first, but to be that it would need to factor to (x+3)(x+3); (x+3)(x+3) FOILed is x^2+6x+9, not x^2+9.
  Button navigates to signup page
  (12 votes)
Lexi Tischman
Posted 6 years ago. Direct link to Lexi Tischman's post “can there be no way to fa...”
can there be no way to factor a problem?
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(5 votes)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “Yes some problems are un-...”
  Yes some problems are un-factorable in the real domain. If you look at b^2 - 4ac, if this is positive you have 2 factors, if it is 0 you have one factor, and if it is negative, is does not have factors in real domain. The first crosses x axis at two places, the second has the vertex on the x-axis, and the third has the vertex above if open upward or below if open downward and never crosses the x axis.
  Comment on David Severin's post “Yes some problems are un-...”
  (12 votes)
Riley
Posted 10 months ago. Direct link to Riley's post “I'm struggling so hard wi...”
I'm struggling so hard with this. I watched every video, read this whole article, thought I understood everything, spent 5 hours studying and doing all the practices until I got 100% on them, but once I read this article and tried to do the practice questions without just being told which method to use upfront I suddenly just can't do anything beyond factoring out the GCF. I could only get 1/7 practice questions on this article right and I'm so frustrated I just want to cry. Does anyone have advice because I just want to give up.
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- 🐱Meg🐱
  Posted 10 months ago. Direct link to 🐱Meg🐱's post “Don't give up! Just remem...”
  Don't give up! Just remember that once you have tried factoring out the GCF, u need to check these things:
  
  1. Is there a difference of squares?
  
  2. Is there a perfect square trinomial?
  
  3. a.) Is there an expression of the form x^2 + bx + c?
  b.) If so, R there factors of c that sum to b?
  
  4. If you've gotten this far, the quadratic expression must be of the form ax^2 + bx + c where a≠1. If there are factors of ac that add up to b, factor using the grouping method.
  
  If there is anything specific u don't understand then I recommend going to previous videos and watching them thoroughly - they should help and I often go back to them bc it is hard to keep up with everything and remember it all.
  Just remember that the five question method in this article works for ANY quadratic expressions.
  If u need any more help u can ask me :D
  
  Hope this helps.
  Comment on 🐱Meg🐱's post “Don't give up! Just remem...”
  (6 votes)
601044
Posted 3 years ago. Direct link to 601044's post “Is a 4th dimensional grap...”
Is a 4th dimensional graph a thing?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Silenyx: YT
  Posted 3 years ago. Direct link to Silenyx: YT's post “It is, but very hard too ...”
  It is, but very hard too graph and not encountered that much (if at all) in real life.
  Button navigates to signup page
  (2 votes)
TheSnowyOwl9455
Posted 10 months ago. Direct link to TheSnowyOwl9455's post “So, question 6 says to fa...”
So, question 6 says to factor 56-18x+x^2 completely. After placing in standard form: x^2 -18X +56, the next step should be to find the GCF. 2( x^2 -9x + 28). After this, there is no a+b=-9 and a * b = 28. Factors of 28 are 1,28, 2, 14, 4, 7. So none of them work. However, apparently, when I asked for the help, they just skipped right past the GCF that they had taught us to try first and went right on to the sum/product step. Does anyone else feel confused about this?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted 10 months ago. Direct link to Kim Seidel's post “The polynomial has no com...”
  The polynomial has no common factor other than 1. In order for there to have been a common factor of 2, the problem would have been: 2x^2-18x+56.
  
  Yes, you should always look for a GCF. But all terms need to be evenly divisible by the value you pick. x^2 does not divide evenly by 2 in your problem, so the GCF=1 and there is no need to factor out a GCF.
  
  Hope this help.s
  Button navigates to signup page
  (8 votes)
Prophark
Posted 4 months ago. Direct link to Prophark's post “Just to confirm Is (x-4)...”
Just to confirm

Is (x-4)^2 = (4-x)^2
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Quantum_Cat
  Posted 4 months ago. Direct link to Quantum_Cat's post “Yes, you are correct here...”
  Yes, you are correct here is why:
  
  (x-4)^2 = x^2 + 2(x)(-4) + 16 = x^2 - 8x + 16
  (4-x)^2 = x^2 + 2(-x)(4) + 16 = x^2 - 8x + 16
  
  Notice, however, that in the second expression it is -x and 4 while in the first it is x and -4.
  Button navigates to signup page
  (4 votes)
Soshi_Garelik
Posted 4 years ago. Direct link to Soshi_Garelik's post “is there any classes that...”
is there any classes that explain this minus the exponents?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Kim Seidel
  Posted 4 years ago. Direct link to Kim Seidel's post “Quadratics have an expone...”
  Quadratics have an exponent of 2. So, I'm not aware of any lessons what would explain it with out using an exponent.
  Button navigates to signup page
  (1 vote)
Peter Leo
Posted 8 months ago. Direct link to Peter Leo's post “I noticed that the formul...”
I noticed that the formula x^2 + (a+b)x + ab doesn't make an appearance here from the Factoring simple quadratics review article but formula x^2 + 2abx + b^2 does. I'm studying for the unit test in the Algebra basics course but cannot figure out which formula applies when factoring. Does it have to do with when the x terms and constant term aren't perfect squares or when the x^2 term is not equal to 1? Thanks in advance.
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- TheReal3A
  Posted 8 months ago. Direct link to TheReal3A's post “`x^2 (a+b)x + ab` is the ...”
  x^2 (a+b)x + ab is the general rule for factorising, basically signifying that you find 2 numbers a and b which add to the coefficient of x and multiply to the constant term, to factorise to (x + a)(x + b) = 0.
  
  x^2 + 2bx + b^2 (not x^2 + 2abx + b^2) is the expansion of (x + b)^2, a perfect square; this is a certain situation in factorising where it's a perfect square. It is useful to know for factoring but is most important in the completing the square method.
  Comment on TheReal3A's post “`x^2 (a+b)x + ab` is the ...”
  (3 votes)
Ella
Posted 7 years ago. Direct link to Ella's post “Is there a perfect square...”
Is there a perfect square trinomial?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “Yes... there is a perfect...”
  Yes... there is a perfect square trinomial. You can find lessons on it at these links:
  1) Multiplying to create a perfect square trinomial:
  -- https://www.khanacademy.org/math/algebra-home/alg-polynomials/alg-special-products-of-polynomials/v/pattern-for-squaring-simple-binomials
  -- https://www.khanacademy.org/math/algebra-home/alg-polynomials/alg-special-products-of-polynomials/v/square-a-binomial
  2) Factoring a perfect square trinomial: https://www.khanacademy.org/math/algebra-home/alg-polynomials/alg-factoring-quadratics-perfect-squares/v/factoring-perfect-square-trinomials
  Comment on Kim Seidel's post “Yes... there is a perfect...”
  (1 vote)

Integrated math 2

Course: Integrated math 2 > Unit 2

What you need to know for this lesson

What you will learn in this lesson

Intro: Review of factorization methods

Putting it all together

Factoring quadratic expressions

Example 1: Factoring 5x2−80‍

Example 2: Factoring 4x2+12x+9‍

Example 3: Factoring 12x−63+3x2‍

Example 4: Factoring 4x2+18x−10‍

Check your understanding

Want to join the conversation?

Example 1: Factoring $5 x^{2} - 80$ ‍

Example 2: Factoring $4 x^{2} + 12 x + 9$ ‍

Example 3: Factoring $12 x - 63 + 3 x^{2}$ ‍

Example 4: Factoring $4 x^{2} + 18 x - 10$ ‍