Part (a) Factorise 𝑥 squared plus
six 𝑥 plus nine.
This is a quadratic expression as
the highest power of 𝑥 is two. To factorise this expression means
to put it into brackets which we’ll multiply together to give the original
As the coefficient of 𝑥 squared —
that means the number in front of 𝑥 squared — is just one, although we haven’t
written it, then this means that the first term in each bracket is just 𝑥 as when
we multiply 𝑥 by 𝑥, we get 𝑥 squared.
We then need to work out what
numbers go inside each bracket so that we get the original quadratic when we
expand. When we have a quadratic of this
form, we recall that the two numbers we’re looking for need to sum or add to the
coefficient of 𝑥. So that’s six in this case. And they need to multiply to the
constant term, which in this case is nine.
We can find this pair of numbers by
first listing the factors of nine. We have one and nine or three and
three. Three and three also add to
six. So these are the two numbers that
we’re looking for. We, therefore, write plus three in
each bracket. And our quadratic has been
As our two brackets are identical,
we have 𝑥 plus three multiplied by itself, 𝑥 plus three. We can also write this as 𝑥 plus
three squared. We can give our answer in either of
these two forms.
Part (b) of the question says,
“Solve eight 𝑦 minus two is greater than or equal to two 𝑦 plus 10.”
Notice that here we’re being asked
to solve an inequality. I read this sign as greater than or
equal to. You can remember which way around
the inequality sign go by remembering that the larger part of the sign is next to
the largest side of the inequality or that the arrow points towards the smaller
Solving an inequality is a lot like
solving an equation. But instead of getting just one or
perhaps two values for our answer, we instead get a range of values. Just like when we solve an
equation, we need to make sure that we perform the same operation to each side of
We notice first that we have terms
involving 𝑦 on each side. So our first step is going to be to
collect them on the same side. It’s always easier to collect terms
on the side which has the larger number of that variable to start with. So we’re going to collect on the
left of the inequality.
To do so, we need to subtract two
𝑦 from each side. On the left, we now have six 𝑦
because that’s eight 𝑦 minus two 𝑦 minus two. And on the right, we just have
10. So our inequality has become six 𝑦
minus two is greater than or equal to 10.
Our next step is to add two to each
side. This cancels the negative two on
the left. And on the right, we now have
12. So we have six 𝑦 is greater than
or equal to 12.
To solve for 𝑦, our final step is
to divide both sides by six as six 𝑦 divided by six gives 𝑦. On the right-hand side, 12 divided
by six gives two. So the solution to our inequality
is 𝑦 is greater than or equal to two.
This includes all of the values
from two upwards. So it would include integer values
like five. But it would also include decimal
values like 8.2 or even irrational numbers like 𝜋.
Now, just one brief word about
solving inequalities. I said that they were very similar
to solving equations. And that’s true. But there is one key
difference. And this occurs if we multiply or
divide both sides of an inequality by a negative value.
If we do this, then we must
remember to reverse the inequality sign. We can see this if we think of the
simple inequality two is less than 10. If we multiplied both sides of this
inequality by negative one but didn’t change the direction of the sign, then we
would have the statement negative two is less than negative 10.
However, this is completely
untrue. Negative two is actually greater
than negative 10 because it’s less far below zero. So when we multiplied by negative
one, we also needed to reverse the direction of the inequality.
Although that wasn’t required in
this question, it’s really important that you remember this when solving