Factorise 𝑥 squared minus 25.
To factorise an algebraic
expression means to write it as the product of two or more factors, which are
multiplied to give the original expression. This expression is a quadratic
expression as it has an 𝑥 squared term and no higher powers of 𝑥. It doesn’t, however, have an 𝑥
term and it’s actually a special type of quadratic expression.
The key to answering this question
is to spot that 25 is actually a square number. It’s equal to five squared. So we can write 𝑥 squared minus 25
as 𝑥 squared minus five squared. Now, the reason this helps is
because an expression that can be written in exactly this form is called a
difference of two squares, which we sometimes see abbreviated to D O T S or
DOTS. This just means that we’re
subtracting one square from another.
A difference of two squares always
factorises in the same way. If we have 𝑎 squared minus 𝑏
squared, then this factorises as 𝑎 plus 𝑏 multiplied by 𝑎 minus 𝑏. So the two brackets are almost
identical. But one has a positive sign between
the terms and one has a negative sign.
We can check this is indeed the
correct factorisation of 𝑎 squared minus 𝑏 squared by expanding our brackets out
again using the FOIL method. The F stands for first. So we need to multiply the first
term in each bracket together. We have 𝑎 multiplied by 𝑎 which
is equal to 𝑎 squared. The O stands for outers. So we’re multiplying the terms on
the outside together. That’s the 𝑎 in the first bracket
and the negative 𝑏 in the second, which gives negative 𝑎𝑏.
Next, I stands for inners. So we’re multiplying the terms on
the inside of this expansion together. So that’s the 𝑏 from the first
bracket and the 𝑎 from the second, which gives 𝑎𝑏. Finally, L stands for last. So we’re multiplying the 𝑏 in the
first bracket with the negative 𝑏 in the second, which gives negative 𝑏
Now, here is the key point. The two terms in the center of our
expansion are identical, apart from the fact they have different signs. We have negative 𝑎𝑏 plus
𝑎𝑏. And so these two terms cancel each
other out directly. And that’s due to the fact that the
two brackets were very similar, but just had different signs.
So the two terms in the center
cancel each other out. And we’ve just left to the 𝑎
squared minus 𝑏 squared, which confirms that our factorisation is correct.
So now, if we apply this rule to
the expression that we’re asked to factorise, where we don’t have 𝑎 squared minus
𝑏 squared, but 𝑥 squared minus five squared, well the 𝑥 is like the 𝑎 and the
five is like the 𝑏. So our first bracket is 𝑥 plus
five and our second bracket is 𝑥 minus five.
You could expand these brackets out
if you wanted to check that the factorisation is correct. And what you’d get is 𝑥 squared
minus five 𝑥 plus five 𝑥 minus 25. And as before, you’d see that those
two terms in the middle would cancel each other out, just leaving us with 𝑥 squared
minus 25. So our factorisation is
So whenever you’re asked to
factorise a quadratic expression, which is just a squared term minus a number, check
whether that number is a square number because if it is, it’s a difference of two
squares. And we can factorise the expression
easily using the method that we have here.