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Worked example: completing the square (leading coefficient ≠ 1)

Sal solves the equation 4x^2+40x-300=0 by completing the square. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user VANESSA FUNG
    The process doesn't seem that confusing but if the result is not a perfect square than what's next? an example is x squared +16x+57=0 the result I go was x+8=7 but this doesn't check... I need help please
    (25 votes)
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    • blobby green style avatar for user MitchellS2394
      1.Subtract 57 from both sides, which will give you x squared+16x=-57.2.Complete the square: x squared+16x+64=7.3.Factor the left side of the equation: (x+8)squared=7.4.Square root both sides of the equation: x+8= positive or negative square root of 7.5. Split the equations into: x+8=positive square root of 7 and x+8=negative square root of 76. Solve both equation!
      (29 votes)
  • old spice man green style avatar for user Jonathan Ziesmer
    Where did the term "coefficient" come from?
    (18 votes)
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  • duskpin ultimate style avatar for user Sarena Kathryn
    So the practice after this video only managed to completely confuse me. Sometimes you divided everything by the leading coefficient, sometimes you don't divide the last term by the leading coefficient, sometimes you multiple the squared middle term by the leading coefficient. The explanations suck as to why you do this and not that, so can someone help me out please?

    Example:
    2x^2 + 3x - 2 = 0
    My process was this:
    2 (x^2 +3/2x - 1)

    Then divide the middle term to get 3/4, then I subtract that term squared from -1 to get -1 - 9/16, to which I got 25/16 = (x+3/4)^2 or 2(x+3/4)^2 - 25/16

    But the hint for the equation showed this process instead:
    2(x^2+3/2x) - 2
    (x + 3/4)^2 - 2 - 2* 9/16
    2(x+3/4)^2 - 25/8

    Why didn't they divide the 2 term by 2 in the beginning? And why did they times the added term by 2 at the end? Looking back at it, I'm thinking they multiplied the last term by 2 to make it even with the equation in the paratheses, but I've also seen equations when the term isn't multiplied by the leading coeffiecient. Help?
    (19 votes)
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    • aqualine ultimate style avatar for user Kunikuzushi
      5 years late, but I believe I know the answer:
      The example question's answer was supposed to be written in vertex form, and the method they used for doing this was by factoring only the first two terms like so:
      2x² + 3x -2 = y
      2(x² + 3/2x) - 2 = y
      Then to make a perfect square, they added 9/16 in the parenthesis to make 2(x + 3/4)². But to retain the exact value of the equation, 9/16 multiplied by 2 (factored from the parenthesis) must be subtracted.
      2(x + 3/4)² - 2 - 2 * 9/16 = y
      2(x + 3/4)² - 25/8 = y
      The process you tried by making y = 0 also could have worked, but there was an error in the last step.
      25/16 = (x + 3/4)²
      2 * 25/16 = 2(x + 3/4)²
      2(x + 3/4)² - 25/8 = 0
      Hope this helps :)
      (3 votes)
  • leaf green style avatar for user dimitri.slegers
    how can my calculator solve this problem using the quadratic formula? since -b+square root (b²-4ac) would be -40+ square root of (40²-4.4.(-300)) which is equal to -40 + square root of (1600 - 4800). wouldn't that be taking the square root of a negative number? or am I using the wrong order of operations?
    (10 votes)
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  • piceratops ultimate style avatar for user FullyTheoretical
    I didn't factor out the 4 at the beginning and ended up with a funky answer with a square root of 7. I understand how to do it properly and got the same answer of x=5 or -15 as Sal did when I factored out the 4 first, but just don't fully get why. Help? Thanks!
    (4 votes)
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    • mr pink green style avatar for user David Severin
      4x^2 + 40x - 300 = 0 so we have 4x^2 + 40x = 300
      Since (ax +b)^2 = a^2x^2 + 2abx + b^2, that means a = 2, so the middle term is 2 • 2 b = 40, so b = 10 and b^2 = 100
      completing the square (2x + 10)^2 = 300 +100
      (2x + 10)^2 = 400, take the square root of both sides, adding +/- on right
      2x +10 = +/- 20, 2x = 20 -10 or 2x = -20 - 10
      2x = 10 or 2x = -30
      x = 5 or x = -15
      Without factoring, there are a whole lot of places to mess up, probably one of the most common mistakes is getting b incorrect. Find where you messed up.
      (7 votes)
  • starky sapling style avatar for user Makado
    Why the 1st coefficient has to be 1?
    (3 votes)
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  • duskpin tree style avatar for user anisha
    How do you solve it if the middle term doesn't factor by the first term? For example, -4x^2-6x-2. -4 does not factor into -6 but by -2 and when you factor it by -2 you are left with a leading coefficient of 2. Thanks!
    (4 votes)
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  • male robot donald style avatar for user MrRevere
    can we use the quadratic formula instead ?
    (2 votes)
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    • leafers ultimate style avatar for user Alex
      Completing the square is actually how you derive the quadratic formula, so it's good to see where it comes from. It's also sometimes useful to put expressions in that form, but in general it's just good to get comfortable with different kinds of algebraic manipulations.
      (5 votes)
  • blobby green style avatar for user merscjul000
    Can someone please explain this in kindergarten terms for me? I have a hard time remembering these kinds of things and I need help with the steps that you take in order because sometimes the person narrating these get a little off course so I get lost in the process of where everything is going. Thank you
    (3 votes)
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  • blobby blue style avatar for user ✨ Sofia Utama 💯
    What is a complete square/perfect square trinomial? (-)
    Also, why a perfect square if we take half of the first degree term and square it? Is it because of 2ax, and a²? Since you have 2ax, you need to divide 10x by 2 to get a, and square a to get the constant?
    What's interesting is that when you factor the x²+2ax+a², you get (x+5)², which uses the 5 from the a. But I guess that makes sense because when you factor, you use (x+a)(x+b).
    (2 votes)
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Video transcript

We're asked to complete the square to solve 4x squared plus 40x minus 300 is equal to 0. So let me just rewrite it. So 4x squared plus 40x minus 300 is equal to 0. So just as a first step here, I don't like having this 4 out front as a coefficient on the x squared term. I'd prefer if that was a 1. So let's just divide both sides of this equation by 4. So let's just divide everything by 4. So this divided by 4, this divided by 4, that divided by 4, and the 0 divided by 4. Just dividing both sides by 4. So this will simplify to x squared plus 10x. And I can obviously do that, because as long as whatever I do to the left hand side, I also do the right hand side, that will make the equality continue to be valid. So that's why I can do that. So 40 divided by 4 is 10x. And then 300 divided by 4 is what? That is 75. Let me verify that. 4 goes into 30 seven times. 7 times 4 is 28. You subtract, you get a remainder of 2. Bring down the 0. 4 goes into 20 five times. 5 times 4 is 20. Subtract zero. So it goes 75 times. This is minus 75 is equal to 0. And right when you look at this, just the way it's written, you might try to factor this in some way. But it's pretty clear this is not a complete square, or this is not a perfect square trinomial. Because if you look at this term right here, this 10, half of this 10 is 5. And 5 squared is not 75. So this is not a perfect square. So what we want to do is somehow turn whatever we have on the left hand side into a perfect square. And I'm going to start out by kind of getting this 75 out of the way. You'll sometimes see it where people leave the 75 on the left hand side. I'm going to put on the right hand side just so it kind of clears things up a little bit. So let's add 75 to both sides to get rid of the 75 from the left hand side of the equation. And so we get x squared plus 10x, and then negative 75 plus 75. Those guys cancel out. And I'm going to leave some space here, because we're going to add something here to complete the square that is equal to 75. So all I did is add 75 to both sides of this equation. Now, in this step, this is really the meat of completing the square. I want to add something to both sides of this equation. I can't add to only one side of the equation. So I want to add something to both sides of this equation so that this left hand side becomes a perfect square. And the way we can do that, and saw this in the last video where we constructed a perfect square trinomial, is that this last term-- or I should say, what we see on the left hand side, not the last term, this expression on the left hand side, it will be a perfect square if we have a constant term that is the square of half of the coefficient on the first degree term. So the coefficient here is 10. Half of 10 is 5. 5 squared is 25. So I'm going to add 25 to the left hand side. And of course, in order to maintain the equality, anything I do the left hand side, I also have to do to the right hand side. And now we see that this is a perfect square. We say, hey, what two numbers if I add them I get 10 and when I multiply them I get 25? Well, that's 5 and 5. So when we factor this, what we see on the left hand side simplifies to, this is x plus 5 squared. x plus 5 times x plus 5. And you can look at the videos on factoring if you find that confusing. Or you could look at the last video on constructing perfect square trinomials. I encourage you to square this and see that you get exactly this. And this will be equal to 75 plus 25, which is equal to 100. And so now we're saying that something squared is equal to 100. So really, this is something right over here-- if I say something squared is equal to 100, that means that that something is one of the square roots of 100. And we know that 100 has two square roots. It has positive 10 and it has negative 10. So we could say that x plus 5, the something that we were squaring, that must be one of the square roots of 100. So that must be equal to the plus or minus square root of 100, or plus or minus 10. Or we could separate it out. We could say that x plus 5 is equal to 10, or x plus 5 is equal to negative 10. On this side right here, I can just subtract 5 from both sides of this equation and I would get-- I'll just write it out. Subtracting 5 from both sides, I get x is equal to 5. And over here, I could subtract 5 from both sides again-- I subtracted 5 in both cases-- subtract 5 again and I can get x is equal to negative 15. So those are my two solutions that I got to solve this equation. We can verify that they actually work, and I'll do that in blue. So let's try with 5. I'll just do one of them. I'll leave the other one for you. I'll leave the other one for you to verify that it works. So 4 times x squared. So 4 times 25 plus 40 times 5 minus 300 needs to be equal to 0. 4 times 25 is 100. 40 times 5 is 200. We're going to subtract that 300. 100 plus 200 minus 300, that definitely equals 0. So x equals 5 worked. And I think you'll find that x equals negative 15 will also work when you substitute it into this right over here.