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Worked example: Rewriting expressions by completing the square

Sal rewrites x²+16x+9 as (x+8)²-55 by completing the square.

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  • blobby green style avatar for user Meghan Lindley
    At , why did you subtract 64 from the other side, where in other videos you added 64? What you do to one side you do to the other?
    (12 votes)
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    • stelly blue style avatar for user Kim Seidel
      Sal doesn't have an equation, so there is no other side. He is working with an expression. To ensure he maintains the original value of the expression, he can only add a value that equal 0 (this is the identify property of addition). This is done by adding the 64 and then also subtracting the 64. 64 - 64 = 0. Thus, he hasn't the expression. He's just making it look different.
      Hope this helps.
      (53 votes)
  • female robot amelia style avatar for user Polo Polo
    what does the b mean? I know it is a constant but, in the equation, what is it?
    (8 votes)
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    • old spice man green style avatar for user Ryulong
      This is what is known as the vertex form of a quadratic equation: it is for finding the vertex (obviously).
      Given the expression (x+a)^2+b, the vertex of this quadratic is (-a,b). So, the b-value is the y-value of the vertex of the quadratic.
      For example:
      (x-7)^2 - 23
      a = -7 and b = -23
      So, the vertex of this quadratic is (-(-7),-23), or (7,-23).
      (16 votes)
  • mr pants pink style avatar for user Anita Buhendwa
    At , how did he geat a=8?
    (4 votes)
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  • blobby blue style avatar for user ✨ Sofia Utama 💯
    At about , Sal adds 64, and subtracts 64. There is no equal sign in the expression, so what he's basically doing is making a zero pair?? Uhhh, why would he do that, again?? And then he factors out the (x+a)² part (that I understand), but when he subtracts the 64 from 9... umm, what's the point of doing that? And isn't the WHOLE THING like, an expression?? Huh. So...why does he have to worry about making the expression true (when he added the 64, he said that he had to subtract a 64) when there's no equal sign?? Very confusing.
    (3 votes)
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    • female robot grace style avatar for user loumast17
      Even though there is no equal sign we are trying to solve this particular expression, or rather find the zeroes of it. Since we want to do it for this particular expression we want to make sure it doesn't change in value, so basically if you graphed x² + 16x + 9, it would be the same graph as x² + 16x + 64 + 9 - 64. Now since these are the same graphs anything we do to the second expression that keeps its value the same will ake it true for the original too.

      Now for that second one if we focus on x² + 16x + 64 hopefully you can see why it turns into (x+8)². Then that just leaves +9 - 64. You could leave that as is and still be right, but there's no reason not to combine like terms.

      Basically if you expanded (x + 8)² - 55 you would get back to x² + 16x + 9, which is the whole point of this. This completeing the square method is just a method to change from one form of quadratics to another.

      Let me kno if something did not make sense though.
      (8 votes)
  • leaf green style avatar for user b
    In the last video it was (x-a)^2 and in this video it is (x+a)^2. How do you decide which one to use?
    (3 votes)
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  • duskpin sapling style avatar for user Lynnet Maravilla
    it's simple, but why choose that formula?
    (3 votes)
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  • spunky sam blue style avatar for user EmmAnueL
    Is completing the square just writing a quadratic in vertex form?
    (1 vote)
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  • starky sapling style avatar for user ✅
    Sal has saved me again
    (4 votes)
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  • male robot hal style avatar for user Aidan Braughler
    Is this supposed to be a way to rewrite quadratic equations into vertex form when I'm trying to graph a parabola?
    (3 votes)
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  • blobby green style avatar for user Nick2022
    Sal calls this technique "Completing the square".

    Is this form called "Perfect Square" form?:


    What do we call this form?:
    x^2 +2ax +a^2 +b

    Is it just called "Completing the square" form?

    I'm trying to memorize these forms so I can recall them easier but not sure what they're named.

    Thank you K.A. and learners
    (1 vote)
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    • mr pink green style avatar for user David Severin
      y=a(x-h)^2+k (similar to your "perfect square" form is actually called vertex form where a is a scale factor and (h,k) is the vertex. Your example just has a=1 and different labels for the vertex which would be at (-a,b). The other two forms are standard y=ax^2+bx+c and factored form y=(ax+b)(cx+d). These are the three forms of quadratics. The idea of a^2x^2 + 2abx + b^2 is just a special case of the standard form which happens to be simple to get into factored form (ax+b)^2. Same with difference of perfect squares where a^2 x^2 - b^2 factors to (ax+b)(ax-b). There are only three forms, there is no perfect square form or completing the square form.
      (5 votes)

Video transcript

- [Voiceover] Let's see if we can take this quadratic expression here, X squared plus 16 X plus nine and write it in this form. You might be saying, hey Sal, why do I even need to worry about this? And one, it is just good algebraic practice to be able to manipulate things, but as we'll see in the future, what we're about to do is called completing the square. It's a really valuable technique for solving quadratics and it's actually the basis for the proof of the quadratic formula which you'll learn in the future. So it's actually a pretty interesting technique. So how do we write this in this form? Well one way to think about is if we expanded this X plus A squared, we know if we square X plus A it would be X squared plus two A X plus A squared, and then you still have that plus B, right over there. So one way to think about it is, let's take this expression, this X squared plus 16 X plus nine, and I'm just gonna write it with a few spaces in it. X squared plus 16 X and then plus nine, just like that. And so, if we say alright, we have an X squared here. We have an X squared here. If we say that two A X is the same thing as that, then what's A going to be? So this is two A times X. Well, that means two A is 16 or that A is equal to 8. And so if I want to have an A squared over here, well if A is eight, I would add an eight squared which would be a 64. Well I can't just add numbers willy nilly to an expression without changing the value of an expression so if don't want to change the value of the expression, I still need to subtract 64. So notice, all that I have done now, is I just took our original expression and I added 64 and I subtracted 64, so I have not changed the value of that expression. But what was valuable about me doing that, is now this first part of the expression, this part right over here, it fits the pattern of a perfect square quadratic right over here. We have X squared plus two A X, where A is 8, plus A squared, 64. Once again, how did I get 64? I took half of the 16 and I squared it to get to the 64. And so the stuff that I just squared off, this is going to be X plus eight squared. X plus eight, squared. Once again I know that because A is eight, A is eight, so this is X plus eight squared, and then all of this business on the right hand side. What is nine minus 64? Well 64 minus nine is 55, so this is going to be negative 55. So minus 55, and we're done. We've written this expression in this form, and what's also called completing the square.