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Basic Quadratic Factoring

Basic Quadratic Factoring

Author: Anthony Varela

Factor a quadratic expression in the form x²+bx+c.

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Hi. This is Anthony Varela and today we're going to practice basic quadratic factoring. So we're going to start off by discussing its connection to FOIL. This is going to be really helpful in understanding why we do what we do when we factor quadratics. We're going to talk about some strategies for basic factoring, and then we're going to look at a couple of examples.

So let's first relate quadratic factoring with FOIL. So when we're FOILing, we have two binomials that we're multiplying together. So here's a more concrete example of FOIL. We have x plus 3 times x plus 2. And what we do first is you multiply those first two terms x times x, so we get x squared. Then we multiply the outside terms, so this would be 2x. Then we multiply the inside terms for 3x, and then we multiply the last two terms, that would give us 6. Now we have two x terms, so we can combine those together, and we have x squared plus 5x plus 6.

Now I want you to note in particular our x term coefficient and our constant term. How do those two numbers relate to 3 and 2? Well, when we add 3 and 2, we get the x term coefficient, but when we multiply 3 and 2, we get our constant term. So when we are factoring quadratics, what we're doing is we're taking something written an expanded form and we're creating an expression in factored form. So we have one factor, x plus 3, being multiplied by another factor, x plus 2. And what we're doing then with quadratic factoring is we really need to decide what are our two integers p and q in this general expression?

And p and q have to sum to equal that x term coefficient, but when you multiply p and q, it has to equal the constant term. So before we get into some examples, let's talk about the sign of pq. So this is that constant term, and this could help you eliminate some choices when you're thinking about two numbers that multiply to the constant term but add to the x term coefficient.

So if we have a positive p and a positive q, that product is going to be positive. If we have a positive p but a negative q, that product will be negative. Same thing if we have a negative p but a positive q. And then if both p and q are negative, our product will be positive. So this can help you eliminate some choices, and you don't have to spend so much time thinking about all the possibilities of what p and q could be. So let's jump into our first example.

We want to factor x squared plus 6x plus 8. So we're going to start with that constant term 8, and let's go ahead and list out a couple of factors of 8. So we're looking for two numbers that multiply to give us 8, and I could refer to my chart here. And notice that we're focusing then on these two squares, so either p and q are both positive, or they could be both negative. Let's start out by listing some positive pairs, and if we don't find what we're looking for we'll consider some negative pairs.

So breaking down 8, well, I could say 8 times 1 equals 8. I could also say 4 times 2 equals 8. And this does it for my positive factors. Let's go ahead and add these two numbers and see if we get 6. Well, 8 plus 1 equals 9, so that's not what we're looking for. 4 plus 2 equals 6. Now this is what we're looking for. So 4 and 2 are our values for p and q.

So writing this out then in factored form, we have x plus 4 and x plus 2. Let's take a look at another example. Here we're introducing a negative number. We have a negative 3 in there, so we're going to start with our constant term, negative 3, and we could see because that constant term is negative, one of our p or q values has to be a negative number. The other one will be positive. So thinking about two numbers that could multiply to negative 3. Well, one possibility is negative 3 times 1. The other possibility is negative 1 times 3.

Now which one of those pairs adds up to positive 2? Well, we can see that it's not this first pair, but it is our second pair. So our value for p is negative 1 and our value for q is positive 3. So in factored form, we have x minus 1 times x plus 3. One final example, x squared minus 6x plus 9. Well, taking a look at our constant term, it is a positive value. So I know that either both p and q are positive or both p and q are negative.

But I see here a negative 6x. Now that tells me that both p and q have to be negative, because I'm not going to have two positive numbers that add to a negative. So let's take our constant 9 and break this down into two negative numbers that could be multiplied together to give us positive nine. We could have negative 3 times negative 3. We could have negative 1 times negative 9. Taking a look at negative 3 plus negative 3, that gives us a negative 6, and that's what we're looking for. Negative 1 plus negative 9 will not give us what we're looking for.

So in factored form, this would be x minus 3 times x minus 3. Here's our p value. Here's our q value. So let's review basic quadratic factoring. To understand basic quadratic factoring, it really helps to understand FOIL, because here we could see that we're looking for two integers, p and q, such that when we add them, we get our x term coefficient, but when we multiply them, we get our constant term. And we also introduced this two by two grid, which could help you look at the sign of that constant term and really eliminate possible p and q values to make that process of identifying p and q just a little bit easier.

So thanks for watching this tutorial on basic quadratic factoring. Hope to see you next time.

Formulas to Know
Basic Quadratic Factoring

open parentheses x plus p close parentheses open parentheses x plus q close parentheses equals x squared plus left parenthesis p plus q right parenthesis x plus p q