We’re now going to take a close look at quadratic polynomials. That is polynomials of degree two. They arise in many problems and applications, so becoming very familiar with them would be an excellent idea. A generic quadratic polynomial, P of x, then is a function of the form ax squared plus bx plus c. The coefficient a has to be different from 0. We’re going to be interested in finding the roots of a quadratic polynomial. And one way to do this is by factoring. Let’s take a equals 1 and consider that we wish to write x squared plus bx plus c in the form of two linear factors. This would be called factoring a quadratic.
So we should be able to write it in the form x plus p times x plus q. What are p and q going to be? Well, here are some clues. When you work out the parentheses, you see that the constant term is pq, which, therefore, will have to be equal to c. And p plus q, the coefficient of x, will have to coincide with b. Now, how do you find these p and q’s? Well, you find them by trial and error, guided by experience. Sometimes you say inspection. These are all fancy ways to say really that we’re going to guess them.
Guessing is not a technique to be scoffed at in mathematics or anywhere else, especially if you learn how to do it in an educated way. Here’s an example. Suppose you wish to factor this quadratic polynomial. Well, we’re going to look for the p and the q among the divisors of 6. And we want two of them that are going to add up to minus 5. So we find rather quickly the factorization x minus 2 times x minus 3. Another example, we look for divisors of the 7. And we find x plus 7 times x minus 1. You see that when you factor the quadratic, its roots become apparent.
In this last example, the two roots are x equals 1 and x equals minus 7. Now, suppose a is different from 1 and the coefficients of the polynomial are integers. We can often still factor by inspection, as we say. Here’s an example. You play around with this. You fiddle around with the possibilities. And pretty quickly you find the factorization that you want for this quadratic. But let’s notice an important fact. Factoring won’t always work. It can’t always work. Why? Because any time you can factor a quadratic into two linear factors, it will have roots. But there are some polynomials of degree two that don’t have roots. Like x squared plus 1, for example, is never equal to 0.
Therefore no matter how hard you try, you’ll never factor it into two linear factors. So this raises a bunch of questions. When is it possible to factor a quadratic? How can we tell? And if it has roots, can we always calculate them by factoring, or is there some other way? It turns out that these general questions can be given a highly satisfactory answer through a technique called completing the square. That means using a little algebra to get a really strong conclusion. Now, we’re going to rewrite our general quadratic polynomial in such a way that just looking at it we’ll be able to tell whether it has roots and what the values of those roots are.
Here’s a simple illustration of the general procedure. Suppose we try to factor x squared plus 2x plus 2. We can spend all the time we want trying to factor it. We won’t manage. Here’s a way of seeing why. We look at the x squared plus 2x, and we have the idea of associating the number plus 1 to it. So we break the 2 into 1 plus 1. That allows us to recognise the first three terms on the right as being x plus 1 squared, by a notable identity that we know very well. And once we’ve written our polynomial in this form, it’s clear that this quadratic has no roots, because the square of something plus 1 can never be 0.
So there are no roots. The polynomial cannot be factored. So this procedure is called completing the square. And now we’re going to do it in a general way in the presence of the symbols a, b, and c. In other words, we’re going to have some fun with algebra. We take our quadratic. And doing a little arithmetic, we write it in a different way and again in a different way. Here, the point of this calculation is to have in the parentheses something that we will recognize as the square of something. And then we have another term outside which we bring into these curly brackets, or braces. And that’s the result we want. We’ll see why in a moment.
Notation, classical since forever. The constants at the end grouped together are called minus delta. And delta, capital delta, is called the discriminant. So definition– delta is b squared minus 4ac, the discriminant of our quadratic polynomial. Let’s summarize what we’ve done. We’ve simply rewritten algebraically our quadratic polynomial in another form. We haven’t really changed anything, except that now it’s going to be apparent that we can prove the following proposition just by looking at this form. For example, the first assertion, if delta is strictly negative, then the quadratic polynomial has no roots. It’s irreducible. Well, that’s obvious.
Because if you look at the quantity in the braces, the minus delta is strictly positive, and you’re adding a square to it, so you can never get 0– no roots. Second case, second assertion, if delta is positive, then there are roots, and they’re given by the formula that you see. That’s easy to prove too because the quadratic will equal 0, and the quantity in braces is equal 0. That means that 2ax plus b squared equals delta. We solve that equation. We get the values of x that we see. Incidentally, this formula for the roots of a quadratic polynomial is called, logically enough, the quadratic formula.
Many students nominate this formula as the formula they’re most likely to remember till the end of their lives. A remark, when delta is 0, we actually get just one root, because the plus or minus doesn’t matter. And otherwise, when delta is strictly positive, we get two distinct roots. Let’s illustrate these results with this example. We want to find the roots of x squared plus x minus 3. Well, here’s the general notation for our quadratic and for the discriminant. You should find a place for this general notation in the software of your mind. After all, you’ll be needing it for the rest of your life. We see that in this example a is 1, b is 1, c is minus 3.
We calculate the discriminant. It’s 13 strictly positive. So there are two distinct roots. We calculate them with the quadratic formula, and there they are.