 A mathematical formula is the automation of a process. So when we automate the process of completing the square to solve an equation, we produce the quadratic formula. Now, since the quadratic formula automates the process of completing the square, you might wonder why it's not called the completing the square formula. And that's because we classify equations by the type of expressions they contain. So remember something like this where we're equating polynomials of degree one is called a linear equation. An equation like this, which deals with a third degree polynomial, is called a third-degree equation. And if the terms of an equation are of the second degree or lower, we should call it a second-degree equation. But people being what they are, we like to have many names for things that are important. So we often call such equations a quadratic equation. So this is a quadratic equation. This is not a quadratic equation because we have this third-degree term, and we would actually call this a third-degree equation. And this, again, is not a quadratic equation because the expressions aren't polynomials. Now the main disadvantage to automating everything and putting it into a formula is that we have to have our equations in standard form. And what this means is that while quadratic equations come in many flavors, for example, we might have x squared plus 3x equals 15, where our constants on one side and our variables are on the other, or we might mix the constant and some variables, or we might separate the quadratic terms from all the rest, or we might have everything on one side equal to zero. While quadratic equations can come in any of these forms, we'll pick one of these and say it's our standard form, and the one we'll pick is this one. We say a quadratic equation is in standard form when it is in the form expression equals zero. So we produce a formula in mathematics by trying to solve a generic equation. So let's try to solve the equation ax squared plus bx plus c, where a, b, and c are any real numbers. Now previously, when we solved equations of this type, we either started with an equation where all our variable terms were on one side and the constant was on the other, or our first step was to rearrange the equation so that the variables were on one side and the constant was on the other. So the suggest the first thing we ought to do is set up our equation so that the variables are on one side and the constant is on the other. So we'll subtract c from both sides. Now another problem we ran into is that if our coefficient of x squared was not equal to one, completing the square was something of a nightmare. So instead, we divided by the coefficient of x squared to make that coefficient equal to one. This made completing the square a lot less nightmarish. So we'll divide everything by the coefficient of x squared. So now let's try to complete the square. Our x term is x squared. Our x term, b over a times x, is 2 times x times b over 2a. And so that means we need to add b over 2a squared to both sides. So we'll do that. Now it'll help to do a little bit of simplification along the way. The right-hand side is minus c over a plus b over 2a squared. So we can simplify that by first of all expanding this b over 2a squared. And now we have a set of fractions. So we can add by finding the common denominator and adding the numerators. And that gives us a slightly more tractable expansion. And we know that the left-hand side must be the square of x plus b over 2a. So now we have square equal expression. So we can take the square root of both sides. Over on the left-hand side, we'll have x plus b over 2a. Over on the right-hand side, we'll have plus or minus a square root. Now we could write that as square root b squared minus 4ac over 4a squared. But let's simplify that a little bit. It's the square root of a quotient. So that'll be the quotient of the square roots. Remember this 4a squared came from 2a quantity squared. So the square root of 4a squared is going to be 2a. And we've already taken care of the plus or minus. So our right-hand simplifies to plus or minus square root b squared minus 4ac over 2a. And finally, we want to solve for x. So let's subtract b over 2a from both sides. And again, we can do a little bit of arithmetic. Here we want to subtract two fractions. Fortunately, they already have the same denominator, so we can just subtract the numerators. And even though we might write this as plus or minus square root minus b, we conventionally move the minus b to the front. And so we get our solution to this generic equation. x equals negative b plus or minus square root b squared minus 4ac, the whole thing over 2a. Now as a general rule, it's not worth memorizing formulas in mathematics. It is far, far, far more important to understand and apply concepts. There's about three exceptions to that rule. The quadratic formula is one of them. So to emphasize that, we'll give it its own page. If I want to solve the equation ax squared plus bx plus c equals zero, then the work that we've just done gives us the quadratic formula that this quadratic equation has the solutions x equals minus b plus or minus square root b squared minus 4ac over 2a. And again, this is one of very few formulas that are actually worth memorizing. Remember this, put it on a t-shirt, tattoo it onto your forehead. Well, don't tattoo it onto your forehead. But it is something that you should keep close to mind. For example, let's take the equation 3x squared plus 7x equals 15. And this is a perfectly good equation that could be solved by completing the square, but let's go ahead and use our quadratic formula. The first important thing is to read all the fine print. The quadratic equation ax squared plus bx plus c equals zero has these solutions, but this is not an equation equal to zero. This is an equation equal to 15. So we'll go ahead and subtract 15 from both sides. And that gives us an equation that we can use the quadratic formula on. So now a, b, and c are the coefficients of the x squared, the x, and the constant term. So we have a equals 3, b equals 7, c equals negative 15. So we'll drop those into the quadratic formula. So every place in the formula we see an a will replace it with 3. Every place in the formula that we see b will replace it with 7. And every place we see c will replace it with negative 15. And if we do the last bits of the arithmetic, we get our solutions using the quadratic formula.