Main content

## Algebra basics

### Course: Algebra basics > Unit 7

Lesson 7: Factoring quadratics: Difference of squares- Factoring difference of squares: leading coefficient ≠ 1
- Factoring quadratics: Difference of squares
- Factoring difference of squares: analyzing factorization
- Factoring difference of squares: missing values
- Factoring difference of squares: shared factors
- Difference of squares

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Factoring difference of squares: shared factors

Sal finds the binomial factor shared by m^2-4m-45 and 6m^2-150.

## Want to join the conversation?

- What do I do in a problem like:

108-3x^2

Where there isn't a perfect square for 108 or 3?

Thank you.(6 votes)- if you factor out the common factor of 3 first, you get 3(36-x^2) which you can factor into 3(6-x)(6+x)(7 votes)

- I didn't understand the video...could someone please explain it again(3 votes)
- We are looking, for a "shared factor" since a factor is, a number/quantity that when multiplied with another produces a given number or expression it should then be shared by both the given quadratic expressions:

m^2-4m-45 and 6m^2-150

The factorization of m^2-4m-45 is = (m+5)(m-9)

And the factorization of 6m^2-150 is: 6(m+5)(m-5)

If you look closely at the factorization of each expression you will see that they share the factor (m+5). What we are looking for is precisely that, the binomial factor they share.(11 votes)

- I can NOT hear this or the previous video.... i have my audio cranked up, no issues elsewhere on the site, can't hear(4 votes)
- you could turn on subtitles.(6 votes)

- Does anyone know how to do this: 2xsquare + 2x = 4

I don't seem to find any videos on these in Khan Academy...

If anyone knows where, please tell me. THANKS!

Please feel free to vote(4 votes)- Notice that the coefficients(the numbers before your x variables) are factor-able by 2. This means you can divide both sides of your expression by two so that you are left with x^2+x=2. You can then subtract 2 from both sides so that you are left with x^2+x-2=0. You can either use the quadratic formula to solve or factor your polynomial into (x+2)(x-1)=0. The solutions are x=-2 and x=1(4 votes)

- Could any help me with this? I'm having a hard time figuring this out.

A man invests $2,400, some at 9.5% annual interest and the balance at 7% annual interest. If he receives $208 in interest, how much did he invest at each rate?

(P.S. If someone knows where these problems are on Khan Academy, let me know.)

Thanks!(3 votes)- Let x = dollars invested at 9.5%

Let y = dollars invested at 7%

$2400 is the total invested, and total tells us to add. This means`x + y = 2400`

Next, I'm assuming you are working with simple interest.

Interest = Amount invested (Percent) (Time).

The 208 is total interest, so again, this means we add.

Interest at 9.5% + Interest at 7% = 208

Use the formula for interest for each component and you get the equation:`0.095x + 0.07y = 208`

You now have a system of linear equations that can be solved with elimination or substitution.

Let's use substitution.

Solve for y in`x+y = 2400`

and you get`y = 2400 -x`

Substitute:`0.095x + 0.07(2400 - x) = 208`

Now solve for "x"

Distribute:`0.095x + 168 - 0.07x = 208`

Simplify:`0.025x + 168 = 208`

Subtract 168:`0.025x = 40`

Divide by 0.025:`x = 1600`

Find "y":`y = 2400 - 1600'`

y = 800`

Thus, $1600 was invested at 9.5% and $800 was invested at 7%

Hope this helps. I haven't seen any videos on this site with problems like this. They have videos on solving systems of equations. You can try an internet search for system of equations problems involving interest.(4 votes)

- Sal has excellent handwriting.(5 votes)
- khan is brilliant , I love you man.(4 votes)
- what does the symbol between m and 2 above mean?(2 votes)
- It's hard to write exponents on a computer. So we use the caret symbol ^

It just means m is being raised to the 2nd power.(4 votes)

- In the exercise below, I tried to solve by stating with -7 like so.

-7(-4 + x^2) = -7(-2^2 + x^2) = then tried to apply difference of squares like so -7(-2+x)(-2-x) which I though was correct but was not. the correct answer was not -7 but 7 and the final result was 7(2+x)(2-x). did I picked the wrong number as -7? I think 7 or -7 should not make any difference!

28−7x^2(2 votes)- You have a sign error going from: -7(-2^2 + x^2) to -7(-2+x)(-2-x). Notice, the binomial has a positive x^2. Your factors create a negative x^2. You could have reversed the 2 terms to: -7(x^2-2^2). Then it's more obvious to see that both x's should be positive in your factors.

Hope this helps.

FYI - It is a good habit to multiply your factors to confirm that they recreate the original polynomial. If they do, then you know the factors are good.(1 vote)

- The first general expression that Sal wrote, (m+a) (m+b), shouldn't it be (m-b)?(2 votes)

## Video transcript

- [Voiceover] We're told that
the quadratic expressions m squared minus 4m minus 45, and 6m squared minus 150, share a common binomial factor. What binomial factor do they share? And like always pause the video and see if you can work through this. All right, now let's work
through this together and the way I am going to do this is I'm just going to try and
factor both of them into the product of binomials and
maybe some other things and see if we have any
common binomial factors. So first let's focus on m squared minus 4m minus 45. So let me write it over here, m squared minus 4m minus 45. So when you're factoring a
quadratic expression like this, where the coefficient
on the, in this case, m squared term, on the
second degree term is one, we could factor it as being equal to m plus a, times m plus b, where a plus b is going to be equal to this coefficient right over here, and a times b is going to be equal to this coefficient right over here. So let's be clear, so, a, let me see another color, so a plus b needs to
be equal to negative 4, a plus b needs to be equal to negative 4, and then a times b needs
to be equal to negative 45. A times b is equal to negative 45. Now I like to focus on the
a times b and think about, well, what could a and b
be to get to negative 45? Well if I'm taking the
product of two things and if the product is negative that means that they are going to have different signs and if when we add them we get a
negative number that means that the negative one has a larger magnitude. So let's think about this a little bit. So a times b is equal to negative 45. So this could be, let's
try some values out. So, 1 and 45, those are too far apart. Let's see. 3 and 15, those still
seem pretty far apart. Let's see, it looks like
5 and 9 seem interesting. So if we say, if we say 5 times, if we were to say, 5 times negative 9, that indeed is equal to negative 45, and 5 plus negative 9 is indeed equal to negative 4. So a could be equal to 5 and b could be equal to negative 9. And so if we were to factor
this, this is going to be m plus 5, times m, I could say m plus negative 9, but I'll just write m minus 9. So just like that I've
been able to factor this first quadratic expression right over there as a product of two binomials. So now let's try to factor the
other quadratic expression. Let's try to factor 6m squared minus 150. And let's see, the first
thing I might want to do is, both 6m squared and 150, they're both divisible by 6. So let me write it this
way, I could write it as, 6m squared minus 6 times, let's see, 6 goes into 150, 25 times. So all I did is I rewrote this and really I just wrote 150 as 6 times 25. And now you can clearly see
that we can factor out a 6. You can view this as undistributing the 6. So this is the same thing as 6 times m squared minus 25, which we recognize this is
a difference of squares. So it's all going to be 6 times, m plus 5, times m minus 5. And so we've factored this out
as a product of binomials and a constant factor here, 6, and so, what is their shared, common or what is their common
binomial factor that they share? Well you see when we factored
it out, they both have an m plus 5. So m plus 5 is the binomial factor that they share.