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# Worked example: Rewriting & solving equations by completing the square

## Video transcript

so let's see if we can solve this quadratic equation right over here x squared minus 2x minus 8 is equal to zero and actually they're cutting down some trees outside so my apologies if you hear some some chopping of trees well I'll try to ignore it myself alright so back back back back to the the problem at hand and there's actually several ways that you could attack this problem you could just try to factor the left-hand side and go that way but the way we're going to tackle it is by completing the square and what does that mean well that means that I want to write I want to write the left-hand side of this equation in into the form X plus a squared plus B and as we'll see if we can write the left-hand in this form then we can actually solve it in a pretty straightforward way so let's see if we could do that let's just remind ourselves what how we need to rearrange the left-hand side in order to get to this form if I were to expand out X plus a squared let me do that in a different color so if I were to expand out X plus a squared that is x squared plus 2ax make sure that plus sign you can see plus 2ax plus a squared and of course you still have that plus B there plus B so let's see if we can write this in that form so what I'm going to do and this is what you typically do when you try to complete the square I'll write the x squared minus 2x now I'm going to have a little bit of a gap and I'm going to have minus 8 and I have another little bit of a gap and I'm going to say equals zero so I just rewrote this equation but I gave myself some space so I can add or subtract some things that might make it a little bit easier to get into this form so if we just matte if we just match our terms x squared x squared 2a X negative 2x so if this is 2a X that means that 2a is negative 2 2a is equal to negative 2 or a is equal to negative 1 another way to think about it your a is going to be half of your first degree coefficient so though our the coefficient on the x-term so the coefficient of the X term is a negative 2 half of that is a negative one and then we want to have and then we want to have an a squared so if a is negative one a squared would be plus one so let's throw a plus one there but like we've done said before we can't just willy-nilly add something on one side of equation without adding it to the other or without subtracting it again on the same side otherwise you're you're fundamentally changing your fundamentally changing the truth of the equation so if I add one on that side I either have to add one on the if I add one on the left side I either have to add one on the right side to make the equation still hold true or I could add one and subtract one on from the left hand side so I'm not really changing the value of the left-hand side all I've done is added one and subtracted one from the left hand side now why did I do this again well now I've been able I haven't changed its its value I just added and subtracted the same thing but this part of the left-hand side now matches this pattern right over here it's x squared plus 2 ax where a is negative 1 so it's minus 2 X plus a squared plus negative 1 squared and then this this part right over here is the plus B so we already know that B is equal to negative 9 negative 8 minus 1 is negative 9 and so that's going to be our B right over there and so we can rewrite this as what I squared off in green that's going to be X plus a squared so we could write it as X plus and I could write a is negative 1 actually let me I could write it like that first X plus a squared or X plus negative 1 well that's just X minus 1 so I'm just going to write it as X minus negative 1 squared and then we have minus 9 minus 9 is equal to 0 is equal to 0 and then I can add 9 to both sides so I just have this squared expression on the left hand side so let's do that let me add 9 to both sides and what I am going to be left with so let me just on the left-hand side those cancel out that's why I added the nine I'm just going to be left with the X minus one squared it's going to be equal to its going to be equal to on this side square plus nine is nine so if X minus one let me do that blue color so it's going to be nine and so if X minus one squared is nine if I have something squared is equal to nine that means that that's something is either going to be the positive or the negative square root of nine so it's going to going to be positive or negative three so we can say X minus one is equal to positive three or X minus one is equal to negative three and you could see that here if X minus one is three three squared is nine if X minus one is negative three negative three squared is nine and so here we can just add one to both sides of this equation add one to both sides of this equation and you get X is equal to 4 or X is if we add one to both sides of this equation we get my digital link is acting up I don't know all right then we get X is equal to negative three plus one is negative two so X could be equal to 4 or X could be equal to negative two and we're done now some of you might be saying well why did we go through the trouble of completing the square I might have been able to just factor this and then solve it that way and and you could have actually for this particular problem completing the square is very powerful because you could actually always apply this and what you will in the future will you will learn the quadratic formula and the quadratic formula actually comes directly out of completing the square and like when you're applying the quadratic formula you're essentially applying the result of completing the square so hopefully you found that fun