Factorise 𝑥 squared minus 𝑥 minus six.
When we factorise an expression, we put it back into brackets. The key clue here on how to do this is actually hidden in the question; it’s this word factor. We begin by looking for a common factor in our terms. If we can find a common factor, we know that we can factorise our expression into one bracket.
Now, actually, 𝑥 squared and negative 𝑥 do have a common factor; it’s 𝑥. But there is no common factor for 𝑥 squared, negative 𝑥, and negative six. So this is a hint that we need to factorise our expression into two brackets.
Now, we can spot fairly quickly what we need to go at the front of each bracket. When we multiply this expression back out again, the product of the first two terms — that’s what you get when you multiply the first two parts — will be 𝑥 squared. So we need an 𝑥 at the front of each bracket since 𝑥 multiplied by 𝑥 is 𝑥 squared.
To find the other bit, we want two numbers that multiply to make negative six and add to make negative one. The adding part is the coefficient of 𝑥, the number of 𝑥s we have. Since there’s no number just, simply a negative symbol, we know that there must be negative one lots of 𝑥.
Now, we will actually for a moment forget about the signs and just begin by finding the factors of six, the numbers that multiply to make six. They are one and six and two and three. We need to find a way to add these numbers to make negative one. Now, there is a way that we can use two and three to get negative one. To get negative one, we’d need to add two and negative three. Two plus negative three is negative one.
We can check that these numbers do indeed multiply to make negative six. We know that two multiplied by three is six and a negative multiplied by a positive is a negative. So two multiplied by negative three is negative six.
We can put these numbers in the brackets in any order. When we factorise 𝑥 squared minus 𝑥 minus six, we get 𝑥 plus two and 𝑥 minus three. Now, it’s always sensible to check your answer by multiplying it back out again using whatever method you are used to.
Let’s check it this time using the grid method. 𝑥 multiplied by 𝑥 is 𝑥 squared, 𝑥 multiplied by negative three is negative three 𝑥, 𝑥 multiplied by positive two is two 𝑥, and two multiplied by negative three — we already said — was negative six.
We should simplify by collecting like terms. Negative three plus two is negative one. So 𝑥 squared minus three 𝑥 plus two 𝑥 minus six simplifies to 𝑥 squared minus 𝑥 minus six as required.
Our expression factorises to 𝑥 plus two 𝑥 minus three.