Why you can't divide by a variable when solving quadratics

0:00- [Narrator] Let's say we wanna solve the equation X squared minus 6X is equal to zero. Like always, pause this video and see if you could solve this on your own before we solve it together. Okay, so first, I will do what you might have been tempted to do. Maybe you say, "Well, let me add 6X to both sides of this equation."

0:23So if I add 6X to both sides, I get on the left hand side, I just get an X squared, and on the right hand side, I get 6X. And then you might be tempted to say, "All right, well, maybe I could divide both sides by X," and you could. So if I divide both sides by X, we have to be careful though.

0:44When you're dividing both sides by X, you have to assume that X cannot be equal to zero. So X cannot be equal to zero, and in order to do this, 'cause if you divided by an X that is zero, then this would be undefined, but if X is not equal to zero, you could keep moving X squared divided by X would be X, 6X divided by X would be six, and you would get X equals six, and this would not be incorrect.

1:09If you would substitute X equals six back into the original equation, you would get six squared minus six times six is equal to zero, 36 minus 36 is indeed equal to zero, but the main question is, did we get all of the solutions? With linear equations, you're used to having one solution.

1:29It turns out with quadratics, second-degree equations like this, you actually can have more than one solution. Now, what could be another solution? Well, we had to assume here that X could not be equal to zero, but what if X was zero? Well, just trying that out, we get zero squared minus six times zero equals zero, and this is true, six times zero is zero, and that is also true.

1:56So X equaling zero is also a solution, but how could we have found that? Well, the correct way when you're dealing with quadratics is not to try to separate the Xs and put some of the Xs on one side or the other and manipulate it that way. It's typically to just try to factor the quadratic expression.

2:18So if we took this original expression, X squared minus 6X, we can factor out an X, so we could say this is the same thing as X times X minus six is equal to zero. Verify this, if you were to distribute this, X times X is X squared, X times -6 is -6X. Now why is this useful?

2:38Well, now, we can see that I have the product of two things. I have the product of X and have the product of X minus six. If I multiply two numbers and I get zero, that means that either of them equaling zero would make the equation true. So I could say either X could be equal to zero or X minus six is equal to zero, and X equals zero is indeed a solution and X minus six equaling zero, that would get us to X equals six.

3:06And we can actually plot the graph of Y equals X squared minus 6X here to see those two solutions, those two places where it equals zero. So this is the graph of Y is equal to X squared minus 6X. And you can see the two places it equals zero are at X equals six and X equals zero, and hopefully this gives you an appreciation for why you can have two solutions here, because it curves, it's a parabola, and it can intersect the X axis twice.