 So here's another thing that's necessary for success in calculus and later mathematics, which is successfully applying the quadratic formula. And this often shows up in problems like the following. We want to solve the equation 12 minus x squared is equal to 4x. And the first thing to notice here is that this equation has an x squared in it. And so we can apply the quadratic formula in general if I have any equation of the form ax squared plus bx plus c equal to zero. I can write down the quadratic formula and by substituting in the correct values of a, b, and c, I can get the solution. And the thing to remember about formulas and theorems and mathematics is, in many ways, they are like legal contracts. As long as you meet certain obligations, then the result will follow. And in this particular case, the obligation is your equation has to be in the form ax squared plus bx equals c. And this is not. This is in a different form. So our very first step, we have to get our equation into the necessary form. And that means that we have to do a little bit of algebra. So here, all of our terms are on one side of the equation. We have zero on the other, so let's move that 4x over. And we're not quite in the right form. Our quadratic formula requires us to have the equation in the form ax squared plus bx plus c. So I'll move the x squared term towards the beginning. It also requires that all of our terms be added. And well, that just means we have to change our coefficients slightly. Instead of negative x squared, we'll have minus 1x squared. Instead of minus 4, we'll use the rule of integer arithmetic that says that the same as plus a negative. And plus 12, we're good on that. So we've met our part of the obligation, which is we have to have the equation in this form. And the mathematics says it is guaranteed that our solution is going to be given by this formula. We just need to identify what a, b, and c are. So let's identify a. a is the coefficient of x squared, which is negative 1. So we have that. b, that's our coefficient of x. We have that, that's negative 4. And then c is the thing that we're adding that is not an x term, not an x squared term, that's our positive 12. So we have a equal to negative 1, b equal to negative 4, and c equal to 12. And I can substitute those into the quadratic formula to get my solutions. So I have my quadratic formula negative b plus or minus b squared minus 4ac all over 2a. So I'll substitute those in. Here's b is negative 4, twice. a is negative 1, twice. And c just appears in this one location. And at this point, it's a matter of doing the arithmetic. Negative negative 4 is positive 4, negative 4 square to 16, 4 times negative 1 times 12. The whole thing subtracted gives us plus 48. 2 times negative 1 is negative 2. And I'll do the addition there. 64 is something I can take a square root of. So I'll go ahead and do that. And one useful thing to keep in mind here is that this plus or minus part of the quadratic formula says to do both. So our solution, we are going to take 4 plus 8 over negative 2. And we're also going to take 4 minus 8 over negative 2. And so again, we'll do the arithmetic there. 4 plus 8 is 12. 4 minus 8, negative 4. And dividing both by negative 2, we get our solutions x equals negative 6 or x equals 2. And those are our two solutions.