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Factoring perfect squares

Sal factors 25x^2-30x+9 as (5x-3)^2 or as (-5x+3)^2. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Mike Haag
    can't seem to solve this problem by grouping as Sal showed on the previous video. This is what I did:

    25y squared - 30x + 9
    25 * 9 = 225
    Factors of 225:
    1, 225
    3, 75
    5, 45
    9, 25
    15, 16
    none of which whose sum is equal to 30.
    In the last video, Sal stated that you first find the product of the first and last monomials, find the factors of that product whose sum is equal to the middle monomial, and from there you can group. I can't find factors of 225 whose sum equals -30. What am I doing wrong?
    (37 votes)
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    • mr pink red style avatar for user K
      actually 15 * 15 is 225 , not 15*16, so we have a.b = 225 and a+b = -30

      so our factors are -15 and -15 thats why he said its a perfect square, so we have 25x^2-15x-15x+9 we factor 5x(5x-3)-3(5x-3) = (5x-3)(5x-3) = (5x-3)^2
      (68 votes)
  • blobby green style avatar for user Megan Devin
    at how did he get the formula (ax+b) squared? How do you know when or when nt to use that formula?
    (17 votes)
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    • winston default style avatar for user Michael Wong
      He wants to show us what would happen if we did (ax + b)^2. He could have really used anything though. But the reasons that he chose this one are...

      1) He knows that if he squares a binomial, he will get a trinomial (which is what he has in the video). So if he wants the most typical binomial squared, he would use (a+b)^2
      2) But since the leading coefficient (the coefficient of the x^2) is not 1, he doesn't want to use (a+b)^2. Instead (ax + b)^2 would work better for this situation.

      So, if you want to know when to use (ax+b)^2, here is the answer:

      If you want to factor a second degree trinomial as a perfect square that doesn't have 1 as the leading coefficient.

      Hope this helps!
      (6 votes)
  • aqualine seed style avatar for user Tarik Caramanico
    Why does it call trinomial?
    (2 votes)
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  • leaf green style avatar for user Nasir Guidry
    Do you have a video over regular factoring of trinomials? ex: 2x squared minus 3x minus 2. Ive been looking everywhere for a video of the sort.
    (6 votes)
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  • primosaur ultimate style avatar for user jean michel henry
    monomial, binomial, trinomial : the terminology goes down to?
    (0 votes)
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  • piceratops tree style avatar for user jesse perdomo
    When factoring your working on making it simpler correct?
    (2 votes)
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    • male robot hal style avatar for user Bollen
      Factoring is the process of turning an expression into a multiplication problem.

      Simplifying is the process of performing all possible operations.

      So these processes actually have opposite results. To help clarify look at the following two example problems, one with the instructions factor, the other with the instructions simplify and look how each starts with the other's answer and ends with the other's question.

      Example 1:
      Simplify 3*5
      Answer: 15

      Example 2:
      Factor 15
      Answer 3*5


      Now for two Algebraic Examples

      Example 3
      Multiply (x+2)(x-3)
      x^2-3x+2x-6
      Answer: x^2-x-6

      Example 4
      Factor x^2-x-6
      (x+?)(x-?)
      (-3)(2)=6 and (-3)+(2)=-1
      Answer: (x+2)(x-3)
      (6 votes)
  • scuttlebug blue style avatar for user sude06
    In the video, Sal showed us 2 possible answers to factor out the trinomial. So, if I have to answer this question like in a test or something, am I supposed to show the 2 possible answers even though they're the same or can I show one of the 2 possible answers for the question to mark right?
    (4 votes)
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    • starky tree style avatar for user keefe
      It depends on what your teacher asks, or how the given prompt is worded. Factoring with the GCF is different, and factoring and polynomial is different.

      Personally, I would ask your teacher for help!
      (1 vote)
  • blobby green style avatar for user Arbaaz Ibrahim
    Does perfect squares just mean that we have two terms that are perfect squares, or does it mean anything else also?
    (3 votes)
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  • spunky sam orange style avatar for user FlamingArrow
    How do you factor something like 4x^2+13x+9 or is it already fully factored?
    (3 votes)
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  • starky ultimate style avatar for user Marko Arezina
    Sorry if this question is off topic but what is the number e?
    (2 votes)
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Video transcript

Factor 25x squared minus 30x plus 9. And we have a leading coefficient that's not a 1, and it doesn't look like there are any common factors. Both 25 and 30 are divisible by 5, but 9 isn't divisible by 5. We could factor this by grouping. But if we look a little bit more carefully here, see something interesting. 25 is a perfect square, and 25x squared is a perfect square. It's the square of 5x. And then nine is also a perfect square. It's the square of 3, or actually, it could be the square of negative 3. This could also be the square of negative 5x. Maybe, just maybe this could be a perfect square. Let's just think about what happens when we take the perfect square of a binomial, especially when the coefficient on the x term is not a 1. If we have ax plus b squared, what will this look like when we expand this into a trinomial? Well, this is the same thing as ax plus b times ax plus b, which is the same thing as ax times ax. Ax times ax is a squared x squared plus ax times b, which is abx plus b times ax, which is another. You You could call it bax or abx, plus b times b, so plus b squared. This is equal to a squared x squared plus-- these two are the same term-- 2abx plus c squared. This is what happens when you square a binomial. Now, this pattern seems to work out pretty good. Let me rewrite our problem right below it. We have 25x squared minus 30x plus 9. If this is a perfect square, then that means that the a squared part right over here is 25. And then that means that the b squared part-- let me do this in a different color-- is 9. That tells us that a could be plus or minus 5 and that b could be plus or minus 3. Now let's see if this gels with this middle term. For this middle term to work out-- I'm trying to look for good colors-- 2ab, this part right over here, needs to be equal to negative 30. Or another way-- let me write it over here-- 2ab needs to be equal to negative 30. Or if we divide both sides by 2, ab needs to be equal to negative 15. That tells us that the product is negative. One has to be positive, and one has to be negative. Now, lucky for us the product of 5 and 3 is 15. If we make one of them positive and one of them negative, we'll get up to negative 15. It looks like things are going to work out. We could select a is equal to positive 5, and b is equal to negative 3. Those would work out to ab being equal to negative 15. Or we could make a is equal to negative 5, and b is equal to positive 3. Either of these will work. If we factor this out, this could be either a is negative-- let's do this first one. It could either be a is 5, b is negative 3. This could either be 5x minus 3 squared. a is 5, b is negative 3. It could be that. Or you could have-- we could switch the signs on the two terms. Or a could be negative 5, and b could be positive 3. Or it could be negative 5x plus 3 squared. Either of these are possible ways to factor this term out here. And you say wait, how does this work out? How can both of these multiply to the same thing? Well, this term, remember, this negative 5x plus 3, we could factor out a negative 1. So this right here is the same thing as negative 1 times 5x minus 3, the whole thing squared. And that's the same thing as negative 1 squared times 5x minus 3 squared. And negative 1 squared is clearly equal to 1. That's why this and this are the same thing. This comes out to the same thing as 5x minus 3 squared, which is the same thing as that over there. Either of these are possible answers.