Proof of the quadratic formula

in the last video I told you that if you had a quadratic equation of the form a x squared plus BX plus C is equal to zero you could use the quadratic formula to find the solutions to this equation and the quadratic formula was X the solutions would be equal to negative b plus or minus the square root of b squared minus 4ac all of that over 2a and we learn how to use it you literally just substitute the numbers a for a B for B C for C and then it gives you two answers because you have a plus or a minus right there what I want to do with this video is actually prove it to you prove that using essentially completing the square I can get from that to that right over there so the first thing I want to do so that I can start completing the square from this point right here is let me rewrite the equation right here so we have a X we do a different color I have a x squared plus BX plus C is equal to zero so first thing I want to do is divide everything by a so I just have a one out here as a coefficient so you divide everything by a you get x squared plus B over a X plus C over a is equal to zero over a which is still just zero now we want to let me get the C over a term onto the right hand side so let's subtract C over a from both sides and we get x squared plus B over a X plus well I'll just leave a blank there because this is gone now we've subtracted it from both sides is equal to negative C over a and I left a space there so that we can complete the square and you saw in the completing the square video you literally just take half of this coefficient right here and you square it so what is B over a divided by two or what is one-half times B over a well that is just B over 2a and of course we are going to square it you take half of this and you squared that's what we do in completing the square so we can turn this into the perfect square into a perfect square of a binomial now of course we cannot just add the B over 2 8 squared to the left hand side we have to add it to both sides so you have a plus B over 2a squared there as well now what happens well this over here this expression right over here this is the exact same thing as X plus B over 2a squared and if you don't believe me I'm going to multiply it out that X plus B over 2a squared is X plus B over 2a times X plus B over 2a x times X is x squared x times B over 2a is plus B over 2a X give B over 2a times X which is another B over 2a X and then you have B over 2a times B over 2a that is plus B over 2a squared that and this are the same thing because these two middle terms B over 2a plus B over 2a that's the same thing as 2 B over 2a X which is the same thing as B over ax so this simplifies to x squared plus B over ax plus B over 2a squared which is exactly what we have written right there that was the whole point of adding this term to both sides so it becomes a perfect square so the left-hand side simplifies to this the right-hand side the right-hand side maybe not quite as simple maybe we'll leave it the way it is right now actually let's simplify it a little bit so the right-hand side we can rewrite it this is going to be equal to if we well this is going to be B squared I'll write that term first this is B let me do it in green so we can follow along so that term right there can be written as B squared over 4a squared and with this term what would that become this would become in order to have 4a squared as the denominator we have to multiply the numerator and the denominator by 4a so this term right here will become minus 4 a c over 4a squared and you can verify for yourself that that is the same thing as that I just multiplied the numerator and the denominator by 4a in fact the 4 is canceled out and then this a cancels out and you just have a C over a so these this and that are equivalent I just switched which I write first and you might already be seeing the beginnings of the quadratic formula here so this I can rewrite this I can rewrite the right hand side right here I can rewrite as b squared minus 4ac all of that over 4a squared this is looking very close to notice b squared minus 4ac it's already appearing that we don't have a square root yet but we haven't taken the square root of both sides of this equation so let's do that so if you take the square root of both sides the left-hand side will just become X plus let me scroll down a little bit X plus B over 2a is going to be equal to the plus or minus square root of this thing it's going to be + the square root of this is the square root of the numerator over the square root of the denominator so it's going to be the plus or minus the square root of b squared minus 4ac over the square root of 4a squared now what is the square root of 4a squared it is to a right to a squared is 4a squared I start 2a squared is 4a squared or the square root of this is that right here so to go from here to here I just took the square root of both sides of this equation now this is looking very close to the quad right we have a b squared minus 4ac over 2 eh now we just essentially have to subtract this B over 2a from both sides of the equation and we're done so let's do that so if you subtract the B over 2a from both sides of this equation what you get you get X is equal to negative B over 2a plus or minus the square root of b squared minus 4ac over 2a common denominator so this is equal to negative B let me do this enough in a new color let's do it in so it's an orange negative B plus or minus the square root of b squared minus 4ac all of that over 2a and we are done by completing the square with just general coefficients in front of our a B and C we were able to derive the quadratic formula just like that hopefully you found that as entertaining as I did