Today, we're going to talk about basic quadratic factoring. So we're going to start by reviewing binomial multiplication, which is basically the reverse process of factoring a quadratic. And then, we'll do some examples of factoring quadratic expressions.
So let's start by reviewing how to multiply these two binomials. We're going to be taking this expression, which is the factored form of a quadratic expression, and writing it in expanded form. And this process is going to be backwards from what we want to do when we factor our quadratics.
So when I'm multiplying these two binomials, I'm going to use FOIL.
I'm going to start by multiplying my first two terms. x times x will give me x squared. Multiplying my outside terms, x and positive 3, will give me positive 3x. Multiplying my inside terms, 2 and x, will give me a positive 2x. And finally, multiplying my last two terms, 2 and 3, will give me 6.
I can combine these two like terms, which will give me the expression x squared plus 5x plus 6, which again is the expanded form of what we started with in factored form.
And what we want to notice is that my two constant terms from my factored form expression can be seen by combining them in some way in my factored form-- or sorry, in my expanded form.
So if I look at the coefficient five of my x term, I see that that's the same as 2 plus 3. And if I look at my constant term, 6, I see that that is the same as 2 times 3. So we want to remember this relationship, because it's going to help us when we are taking our expressions from expanded form into factored form.
So let's look at the general strategy for factoring, which again, is to take an quadratic expression in expanded form and write it in factored form. And we call this "factored form," because x plus p and x plus q are the factors of the quadratic expression.
And so the general strategy of factoring is to find the values of integers p and q so that they satisfy the expanded form, where p plus q will be the value of the coefficient in front of the x-term, and p times q will be the value of the constant term.
So let's look at what's the sign of the constant term means for our integers, p and q. If we were to draw a table and look at the positive and negative values of p, as well as the positive and negative values of q, I know that when I'm looking at my constant term, it's a product of p and q.
So if p then q are both positive, the product will be a positive pq. If p is negative and q is positive, the product will be negative. Similarly, if q is negative and p is positive, the product will be negative. But if p is negative and q is negative, we know that the product will be positive. A negative times a negative is positive.
So when you're factoring and you're looking for values for p and q that satisfy the values for the x-term and the values for the constant with addition and multiplication, you know that if the constant term is going to be positive or if the constant term is positive, then that means our value for p and q must be that either p and q are both positive, or that p and q are both negative.
And again, if your constant term is negative, then we know that it falls into one of these cases, where either p is positive and q is negative, or p is negative and q is positive. In other words, one of the integers is positive and one of them is negative. So let's do some examples.
So let's do some examples factoring quadratic equations. So for my first example, I've got x squared plus 5x plus 4 equals 0. I want to write this expression, which is in expanded form, in factored form. And to do that, I'm looking for two integers that multiply to give me positive 4 and add to give me positive 5.
So I'm going to start by thinking of pairs of integers that multiply to give me positive 4. And since it's a positive number that I'm looking for, I know that both p and q, the two integers, are both either going to be positive or both be negative.
So thinking of my integers that multiply to give me positive 4, I know positive 4 and 1 would work. Negative 4 and negative 1 would work. Positive 2 and positive 2, or negative 2 and negative 2.
So now, I want to see which of these pairs, when the two numbers are added together, will give me a positive 5. Positive 4 plus positive 1 will give me a positive 5.
So my first pair of my first two integers gave me the value that I was looking for, a positive 5. So now, I can write my expression in expanded form, using my two integers of positive 4-- so x plus 4-- and my other integer, positive 1. So x plus 1. And that's equal to 0.
So I wrote this expression, which is an expanded form, in factored form for my quadratic equation.
Let's do another example. I've got x squared plus 2x equals negative 8. So now, I'm looking for two numbers that multiply to give me negative 8 and add to give me a positive 2. Since I'm looking for our two numbers that multiply to give me a negative value, I know that one of the integers will be positive and one will be negative.
So making my list, I could have negative 8 and positive 1, negative 1 and positive 8, negative 2 and positive 4, or negative 4 and positive 2. And again, I'm looking for which of these pairs, when I add the two numbers together, will give me positive 2.
So checking each pair, negative 8 and 1 added will give me a negative 7. Negative 1 and positive 8 added together will give me positive 7, so neither of those will work. But negative 2 plus a positive 4 will give me a positive 2.
So I've identified my two integers. And using them to write my expression in factored form, I will have x minus 2 times x plus 4, and that is equal to 0.
And for my last example, I've got x squared minus 7x plus 10 equals 0. So I'm looking, again, for two integers that multiply to positive 10 and add to negative 7. Again, since they need to multiply to be a positive number, I know that both of the integers will both be positive or both negative.
So I can start by making my list. I can have negative 5 and a negative 2, positive 5 and positive 2, negative 10 and negative 1, or positive 10 and positive 1. So now, I want to see which of those pairs, when I add the two numbers together, will give me a negative 7.
Negative 5 plus a negative 2 gives me negative 7. So right away, I identified my two integers, and I can use them to write in factored form. I'll have x minus 5 times x minus 2 equals 0.
So let's go over our key points from today. Factoring is the process of writing an equation from expanded form to factored form. And factoring involves finding two integers whose sum is the coefficient of the x-term, and whose product is the constant term of the quadratic equation.
So I hope that these key points and examples helped you understand a little bit more about basic quadratic factoring. Keep using your notes and keep on practicing, and soon, you'll be a pro. Thanks for watching.