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Factoring difference of squares: shared factors

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

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  • mr pants teal style avatar for user Emmery
    What do I do in a problem like:
    108-3x^2
    Where there isn't a perfect square for 108 or 3?
    Thank you.
    (5 votes)
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  • old spice man green style avatar for user Tejaswini Swaroop
    I didn't understand the video...could someone please explain it again
    (2 votes)
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    • mr pants teal style avatar for user Rodrigo Segura
      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.
      (8 votes)
  • aqualine seedling style avatar for user Christina
    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)
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    • stelly blue style avatar for user Kim Seidel
      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)
  • aqualine seedling style avatar for user Christina
    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
    (2 votes)
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    • male robot hal style avatar for user Griffin McGrath
      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
      (3 votes)
  • duskpin seed style avatar for user Alan Fellheimer
    what does the symbol between m and 2 above mean?
    (2 votes)
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  • blobby green style avatar for user Arbaaz Ibrahim
    The first general expression that Sal wrote, (m+a) (m+b), shouldn't it be (m-b)?
    (2 votes)
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  • mr pants teal style avatar for user Rebecca
    I just dont get this. At sal says that thay share a comon factor , I do not know what this is.
    (1 vote)
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  • duskpin sapling style avatar for user mluna51
    I have x^4-y^4 and it says to factor completely how do I this can I get help?
    Also how can I request for KA to do video on a subject I want to learn more about?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      You have a difference of 2 squares
      x^4 - y^4 = (x^2)^2 - (y^2)^2
      It factors into: (x^2 - y^2) (x^2 + y^2)
      The 1st binomial is another difference of squares. So, it factors further.
      You get: (x - y) (x + y) (x^2 + y^2)
      Hope this helps.
      (2 votes)
  • blobby green style avatar for user Jay
    when do i use this in life that is not to pass a test? give me a real life example
    (1 vote)
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    • piceratops ultimate style avatar for user Ben Luke
      essentially all forms of advanced math are useless depending on what future career you plan on taking.
      For example, knowing Geometry is good but will prove useless in a field of fiction book writing, but can prove very useful in space travel and weather patterns.

      So basically, you will be taught this stuff, but the likelihood you will ever use it in real life for a future career you take is likely. It's like an emergency kit in case you do need it.
      (1 vote)
  • blobby green style avatar for user Mireille Pimentel
    how do i solve 9m^2 - 15m + 225
    (1 vote)
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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.