Proof of the quadratic formula

In the last video, I told you that if you had a quadratic equation of the form ax 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 learned 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 in 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 ax-- let me do it in a different color-- I have ax squared plus bx, plus c is equal to 0. So the first I want to do is divide everything by a, so I just have a 1 out here as a coefficient. So you divide everything by a, you get x squared plus b over ax, plus c over a, is equal to 0 over a, which is still just 0. Now we want to-- well, let me get the c over a term on to 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 it blank there, because this is gone now; we subtracted it from both sides-- is equal to negative c over a 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 1/2 of this coefficient right here and you square it. So what is b over a divided by 2? Or what is 1/2 times b over a? Well, that is just b over 2a, and, of course, we are going to square it. You take 1/2 of this and you square it. That's what we do in completing a square, so that we can turn this into the perfect square of a binomial. Now, of course, we cannot just add the b over 2a 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 2ax. You have b over 2a times x, which is another b over 2ax, 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 2b over 2ax, 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, 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-- 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 square. And what's 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 4ac 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's cancel 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. Notice, b squared minus 4ac, it's already appearing. 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. And 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 2a, right? 2a squared is 4a squared. 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 quadratic. We have a b squared minus 4ac over 2a, 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 do 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 in a new color. So it's 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.