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
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
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.