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

we're asked to complete the square to solve for x squared plus 40 X minus 300 is equal to zero so let me just rewrite it so 4x squared plus 40 X minus 300 is equal to zero so just as a first step here I don't like having this four out front as a coefficient on the x squared term I 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 divide before 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 if as long as I do I can do any as long as that whatever I do to the left since that I also do to the right hand side that will make the equal the Equality continue to be valid so that's why I can do that so 40 divided by 4 is 10 X and then 300 divided by 4 is what that is 75 let me verify that 400 4 goes into 3 4 goes into 30 7 times 7 times 4 is 28 you subtract you get a remainder to bring down the 0 4 goes into 20 5 times 5 times 4 is 20 subtract 0 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 turn so somehow turn whatever we have on the left hand side into a perfect square and I'm gonna 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 it 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 I'm 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 where 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 we looked at we saw this in the last video where we constructed a perfect square trinomial is that this last term or let's just say this this 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 term a constant term that is the square of half of the coefficient on this 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 to 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 what two numbers if I add them I get 10 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 is that you find that confusing or you could look at the last video on 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 something right over here if I say something squared is equal to hundred 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 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 ten on this side right here I can just subtract five from both sides of this equation and I would get I'll just write it out subtracting five from both sides I get X is equal to five and over here I could subtract five from both sides again I subtracted 5 in both cases subtract five again and I can get X is equal to negative 15 so those are my two solutions 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 let's try with let's try with five I'll just do one of them I can now leave the other one for you I'll leave the other one for you to verify that it works so four four times x squared so four times 25 plus 40 times five plus 40 times five minus 300 needs to be equal to zero four times 25 is 140 times five is 200 we're going to subtract that three hundred one hundred plus 200 - 300 that definitely equals zero so x equals five worked and I think you'll find that x equals negative fifteen will also work when you substitute it into this right over here