Worked example: Completing the square (intro)

use completing the square to find the value of C that makes x squared minus 44 X plus C so we need to figure out a see that makes it a perfect square trinomial in Tri gnome is just a polynomial with three terms here then write the expression as the square of a binomial so we have x squared minus 44 X plus C so how do we make this into a perfect square well if you just look at the traditional pattern for a perfect square let's just think of it in terms of X plus a squared that's the same thing as X plus a times X plus a and we've seen this before and if you were to multiply this out that's x times X which is x squared plus x times a which is ax plus a times X which is ax plus a times a which is a squared so it's x squared plus 2 ax to ax these two you have an ax plus an ax gives you two a X plus a squared so if we can get this into this pattern where I have whatever value is here if I take half of it right this is going to be 2a here if I take half of it and square it over here then this will be a perfect square so if we look over here this thing right here is 2a if we want a pattern match if we want to make this look like a perfect square that has to be 2a so negative 44 is equal to 2a and this right here this C if we pattern match that has to be equal to C has to be equal to a squared so what's a well if we know negative 44 is 2a we can divide both sides of that by 2 divide both sides by 2 and we know that a or we know that negative 22 has got to be equal to a a has got to be equal to negative 22 a is half of the coefficient right here it's half of negative 44 and whenever you complete the square it's always going to be half of the coefficient right here now if that's a what does C need to be well C needs to be a squared in order for this to be a perfect square so C needs to equal C needs to equal negative twenty two squared and we could figure that what that out what that is 22 times 22 we could put the negative later actually going to be the same thing because the negative times a negative is a positive 2 times 22 is 44 put a 0 2 times 22 is 44 get a 4 get an 8 get a 4 so it's 484 so if we were to rewrite this as if we were to rewrite this is x squared minus 44 X plus 484 then this is a perfect square trinomial and or we could write it like this this is x squared minus 2 times minus 2 times or maybe I should write it this way plus 2 times negative 22 X plus negative 22 squared and when you view it that way it's pretty clear that this is a perfect square and then if you were to factor it it's the same thing as X minus 22 times X minus 22 or X minus 22 squared these are all equivalent statements