Factor quadratics with an 𝑥 Squared Coefficient of One
Factoring is the conversion of an expression to an equivalent form with one term, and it is the opposite of the distributive property. So as a reminder, the distributive property is that 𝑎 multiplied by all of 𝑏 plus 𝑐 is equal to 𝑎𝑏 plus 𝑎𝑐. So the opposite of that is factoring, so 𝑎𝑏 plus 𝑎𝑐 is equal to 𝑎 multiplied by all of 𝑏 plus 𝑐, where 𝑎 is the greatest common factor or GCF for short.
So this takes us from two terms to one term which is fully factored now let’s have a go at actually factoring a quadratic. So if we have to factor the quadratic 𝑥 squared plus seven 𝑥 plus twelve. Then we need to remember first is that every quadratic comes in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 where 𝑎, 𝑏, and 𝑐 are all constants. So in this case, we can see that our 𝑎, our coefficient, of the 𝑥 squared is equal to one. So we don’t need to worry about that too much. But to factor a quadratic, the first thing we need to do is find its factors. And to do that, we need to find what they add to get and what they multiply to get. In another words, what’s their sum and what’s their product? So we need a table.
And their sum will always be 𝑏 whereas their product will be 𝑎 multiplied by 𝑐. But as I said a moment ago, we don’t need to worry about it too much in this case because it’s equal to one. So we can see for this one if we draw a table, its sum would be 𝑏, which is seven, and 𝑐 being twelve. So we need to now look through the factor pairs of twelve and see which of those could add to get seven. So first of all, one and twelve, nope that doesn’t add to get seven. So how about two and six? Nope And then finally, three and four.
So yea, three add four is seven, and I’m sure that many of you knew that that factor pair was what we we’re looking for before you even had to do anything. But it’s always good to check through. So because 𝑎 is equal to one, we can just write the brackets with the 𝑥s in there straight away. And we say 𝑥 add three all multiplied by 𝑥 add four.
So this example, we had only positive numbers. For our next two examples, we’re gonna look at how it’s different when we have negative numbers. So we have to factor 𝑥 squared minus four 𝑥 plus three. We know the very first thing we always have to do is say what’s the product and what’s the sum. So what’s it add to get? Well it adds to get negative four, and it multiplies to get three.
So we have two numbers that multiply to give a positive number that add to give a negative, so we know that a negative multiplied by a negative is equal to a positive. So therefore, because they add to give a negative and multiply to give a positive, then we’re gonna be looking for two negative numbers. But anyway, let’s list the only factor pair of three, because it’s prime, and that is one and three.
Well if both of those are negative, they add to give negative four. So it works, so we’re gonna write our brackets out straight away, our parenthesis, and put 𝑥 in them and then our factors. So 𝑥 minus one all multiplied by 𝑥 minus three. There we have it. Done!
Okay so there’s one more type of negative numbers when you’re factoring so let’s look at that so we need to factor 𝑥 squared minus 𝑥 minus thirty. Again we need a table. We say what’s it add to get what’s it times to get. So having a look at the coefficient in front of the 𝑥, kinda looks that there isn’t one. We have to remember that that is hiding a one. So that was negative one 𝑥, so in this case it adds to get our factors; add to get minus one, and then multiply to get negative thirty. So as we have two factors that multiply to give a negative number, that means that one of them must be positive and one of them must be negative, as a negative multiplied by a positive is equal to a negative.
So now listing the factor pairs of thirty, we’ve got one and thirty. Now we’re looking for a difference of one; one and thirty do not have a difference of one so it won’t be them. And then two goes in fifteen times, and they don’t have a difference of one. Three goes in ten times. Again, no difference of one. Four does not go in. But five does, and five goes in six times.
And happily, they have a difference of one. But now we need to work out which one is gonna be negative. So because when they add together they give us a negative number, that means the larger number, or in this case six, has to be negative. So we’re gonna put our parentheses straight down, put the 𝑥s straight in them and then pop in the factors. So we have 𝑥 plus five all multiplied by 𝑥 minus six.
And there we have it. Done! So we must remember when we are factoring quadratics, the very first thing we need to do is do a table. So what’s add to get what’s it times to get? And then once we’ve done that, we need to focus on what are our negatives and positives. So we know negative multiplied by a positive is equal to a negative, and so on and so for. We list the factor of pairs, finally work out which ones are going to be negative if they are, and then you put them straight into parentheses. So we have learned how to factor when the coefficient of 𝑥 squared is equal to one.