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Factoring difference of squares: leading coefficient ≠ 1

Sal factors 45x^2-125 as 5(3x+5)(3x-5). Created by Sal Khan.

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  • ohnoes default style avatar for user Brian Lee
    Can 25 can be squared? Yes or no?
    (25 votes)
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    • aqualine ultimate style avatar for user Mark Timanovsky
      Brian Lee
      25 can be squared, and so can any number. Squared means to multiply by itself. When you are finding a square root, not every number can have a square root. There is a helpful song that i know, ahem:
      1x1=1
      2x2=4
      3x3=9
      4x4=16
      5x5=25********
      6x6=36
      7x7=49
      8x8=64
      9x9=81
      10x10 is a HUNDRED! EVERY BODY DANCE!
      So yes, 25 can be squared and 25 squared will be 625. as i said, any number can be squared. Not every number has a square root. some examples of numbers not have ing a square root is:2,3,5,6,7,8,10, etc. Some number have a square root that is a decimal, but not all. Most do, that arent normally square rotted. Hope this helps :)
      (30 votes)
  • blobby green style avatar for user SometicHD
    How would you solve a difference of squares for 100g^2 -1
    (7 votes)
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  • blobby green style avatar for user theodalemanubay
    what is the purpose of the 5 outside the perenthesis
    (6 votes)
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    • stelly blue style avatar for user Kim Seidel
      The 5 is the Greatest Common Factor that was factored from the polynomial using the distributive property. The new binomial is a difference of 2 squares, so it can be factored to another level using that technique. Thus you create 3 factors: 5 (3x - 5) (3x + 5)
      (13 votes)
  • aqualine ultimate style avatar for user Calvin Haworth
    How do you solve these types of problems when the constant is a non-square number?
    For example (and don't just solve it!):
    96−6x^2
    (5 votes)
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    • leaf green style avatar for user wiiwizard0
      First, you must divide by a common factor until both terms are perfect squares, if possible. In this problem, 96 and -6 are both divisible by 6.
      After the division, we have 16-x^2. Now you can factor it in the "difference of two squares" method. The square root of 16 is 4, and the square root of x^2 is x.
      Important: The factor with the subtraction/negative sign always comes second in the factored form.
      Important: Don't forget the factor you divided by! It's the leading coefficient in your final answer.
      Knowing this, the factored form of 96-6x^2 is 6(4+x)(4-x).

      I hope this cleared things up.
      (7 votes)
  • leaf green style avatar for user Timofte Bogdan
    why dint he considered sqrt(45)x to power of 2 - sqrt(125) to power of 2 ?
    (5 votes)
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  • blobby green style avatar for user Jayanth Samala
    What does ^ mean?
    (1 vote)
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  • blobby green style avatar for user Rustenberg
    How can factor 3x^2 + 10x + 8 in the forn of (3x ) (x )
    or 12x^2 - 17x + 6 = (4x ) (3x )?
    Is there a rule to do this? Completing the square will not do.
    Where on the Academy can I find the videos for this?
    I believe that it is most trial and error, but is a way for trial and least error?
    Very, very,very much thanks.
    Your advice may also be sent to rainer@dds.nl
    Thank you, happy study on the Khan Ac.
    (3 votes)
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  • piceratops ultimate style avatar for user Liam Nickell
    If this were too be an actual question on a test, which answer would be accepted? Or would both be acceptable? 5(9x^2 - 25) or 5(3x + 5)(3x - 5)?
    (4 votes)
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    • piceratops ultimate style avatar for user Just Keith
      That would depend on what the question asked for. If it asks for "fully factored" then 5(3x + 5)(3x - 5) would be correct.

      So, you need to know the names of the various forms for expressions so that you know which form a test question is asking for.
      (3 votes)
  • blobby green style avatar for user Dylan McKinney
    Would (15x+25)(15x-25) be a correct answer due to the distributive property? Would it count as simplying and be a valid answer for a question asking for simplification or would it just be... Unsimplifying again? lol
    (2 votes)
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  • piceratops seed style avatar for user achestnutt14
    How do you factor the difference of two squares?
    (3 votes)
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    • duskpin ultimate style avatar for user Benjamin
      Without variables:
      25 - 9 = 5² - 3² = 5² + 15 - 15 - 3² = (5 + 3)(5 - 3)
      With one variable:
      x² - 9 = x² - 3² = x² + 3x - 3x - 3² = (x + 3)(x - 3)
      With two variables:
      x² - y² = x² + xy - xy - y² = (x + y)(x - y)
      When you distribute the first group onto the second, the combined terms cancel out because the two groups have opposite signs and you're left with the first and second terms of each group squared.
      (2 votes)

Video transcript

Let's see if we can factor the expression 45x squared minus 125. So whenever I see something like this-- I have a second-degree term here, I have a subtraction sign-- my temptation is to look at this as a difference of squares. We've already seen this multiple times. We've already seen that if we have something of the form a squared minus b squared, that this can be factored as a plus b times a minus b. So let's look over here. Well, over here, it's not obvious that this right over here is a perfect square. Neither is it obvious that this right over here is a perfect square. So it's not clear to me that this is a difference of squares. But what is interesting is that both 45 and 125 have some factors in common. And the one that jumps out at me is 5. So let's see if we can factor out a 5, and by doing that, whether we can get something that's a little bit closer to this pattern right over here. So if we factor out a 5, this becomes 5 times-- well, 45x squared divided by 5 is going to be 9x squared. And then 125 divided by 5 is 25. Now, this is interesting. 9x squared-- that's a perfect square. If we call this a squared, then that tells us that a would be equal to 3x. 3x-- the whole thing squared is 9x squared. Similarly-- I can never say similarly correctly-- 25 is clearly just 5 squared. So in this case, if we're looking at this template, b would be equal to 5. So now this is a difference of squares, and we can factor it completely. So we can't forget our 5 out front that we factored out. So it's going to be 5 times a plus b. So let me write this. So it's going to be 5 times a plus b times a minus b. So let me write the b's down, plus b and minus b. And we're done. 5 times 3x plus 5 times 3x minus 5 is 45x squared minus 125 factored out.