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# Quadratic formula (proof)

## Video transcript

in the completing the square video I kept saying that all the quadratic equation is is completing the square is kind of a shortcut of completing the square and I I was under the impression that I had done this proof already but now I realize that I haven't so let me prove the quadratic equation to you by completing the square by completing the square so let's say I have a quadratic equation I guess a quadratic equation is actually what you're trying to solve and what a lot of people call the quadratic equation is actually the quadratic formula but anyway I don't want to get caught up in terminology but let's say that I have a quadratic equation that says let's it's a x squared plus B X plus C is equal to zero and let's just complete the square here so how do we do that well let's divide but let's subtract C from both sides so we get a x squared plus B X is equal to minus C and just like I said in the completing the square video I don't like having this a coefficient here I like to that I like just having a one coefficient on my x squared term so let me divide everything by a so I get x squared plus B over a X is equal to you have to divide both sides by a minus C over a now we are ready to complete the square and what was completing the square well it's somehow adding something to this expression so it has the form of something that is the square of an expression what do I mean by that well you know let me I'll do a little aside here if I told you that x squared you know X plus a squared that equals x squared plus 2a X plus a squared right so if I can add something here so that this left-hand side of this expression looks like this then I will then I could I could go the other way and I could say oh well then this is going to be X plus something squared so what what do I have to add on both sides and if you watch the completing the square oh this should be hopefully intuitive for you what you do is you say well this be B over a this corresponds to the to a term so a is going to be half of this right it's going to be half of this coefficient that would be the a and then the what I need to add is a squared so I need to take half of this and then square it and then add it to both sides and let me do that in a different color under this magenta so I'm gonna take half of this I'm just completing the square that's all I'm doing no magic here so plus half of this well half of that is B over 2a right just multiply by 1/2 and I have to square it well if I did to the left-hand side of the equation I have to do it to the right-hand side so plus B over 2a squared and now I have this left-hand side of the equation in the form that it is the square of an expression that's you know X plus something and what is it well it's that's equal to let me switch colors again what's the left-hand side of this equation equal 2 and you can just use this pattern and go to the left or you know it's X plus what well we said a you could view one of two ways a is half of this coefficient or a is the square root of this coefficient or since this is we didn't even square it we know that this is a B over 2a is a so this is the same thing as X plus B over 2a everything squared and then that equals that equals let's see if we can if we can simplify this or make this a little bit cleaner that equals C if I were to have a common denominator I'm just doing a little bit of algebra here see when I squared this is going to be 4a squared so let me let me write this this is equal to B squared over 4a squared right and so if I have to add these two fractions let me make this equal to four a squared right and then if the denominators for a squared plus a minus C over a become well I would have to see if I'm multiplying the denominator by 4a I have to multiply the numerator by 4a so this becomes minus 4 a c right and then B squared over 4a squared well that's just still be B squared I'm just doing a little bit of algebra hopefully I'm not confusing you I just I expanded this I just took the square of this B squared over 4a squared and then I added this to this I got a common denominator and minus C over a is the same thing as minus 4ac over 4a squared and now we can take the square root of both sides of this equation and this should hopefully start to look a little bit familiar to you now so let's see so we get X so if we take the square root of both sides of this equation we get X plus B over 2a is equal to the square root of this so let's take the square root of the numerator and the denominator so the numerator is I'm going to put the B squared for I'm just going to switch to this order doesn't matter the square root of b squared minus 4ac right that's just the numerator that is just that I just took the square root of it and we have to go to the square root of the denominator 2 what's the square root of 4a squared what's just 2 a right to a and now what do we do oh and and it's very important when we're taking the square root it's not just the positive square root it's the positive or minus square root and we saw that couple of times when we did the you know you could say it's a plus or minus here too but if you're plus or minus on the top and are plus or minus the bottom you can just write it once on the top and I'll let you think about why you only have to write it once right if you had a negative and a plus or a negative and a plastic sometimes cancel out or a negative and a negative that's the same thing as just having a plus on top anyway I think you get that and now we just have to subtract B over 2a from both sides and we get we get and this is the exciting part we get X is equal to minus B over 2a plus or minus this thing so minus B squared minus 4a see all of that over 2a and we already have a common denominator so we can just add the fraction so we get and we're going to do this in a vibrant bold I don't know maybe not so much well the green color so we get X is equal to the numerator negative B plus or minus square root of b squared minus 4a see all of that over 2a and that is the famous quadratic formula so there we go we proved it and we proved it just from completing the square I hope you found that vaguely interesting see you in the next video