Factorise five 𝑥 squared minus seven 𝑥 minus six.
The first thing we’re going to do is take the coefficient from the 𝑥 squared term and the constant negative six and multiply them together. Five times negative six equals negative 30. From there, we need to find a factor pair of negative 30 that sums to negative seven.
Considering all the factor pairs of negative 30, negative one times 30, positive one times negative 30, negative two times 15, two times negative 15, negative three times 10, positive three times negative 10, negative five times six and five times negative six. and we need the pair that sums to negative seven. Three plus negative 10 equals negative seven and three times negative 10 equals negative 30.
We’re now going to take the negative seven 𝑥 and break it up into two terms that would add together to equal negative seven 𝑥. Three 𝑥 minus 10𝑥 equals negative seven or negative 10 𝑥 plus three 𝑥 equals negative seven. Either one of these would work.
Let’s go with three 𝑥 minus 10 𝑥 and then bring down your five 𝑥 squared and negative six. At this point, we want to see what factors five 𝑥 squared and three 𝑥 have in common and what factors negative 10𝑥 minus six have in common. Five 𝑥 squared plus three 𝑥 have a common factor of 𝑥. And if we take that factor out, five 𝑥 squared becomes five 𝑥 and three 𝑥 becomes three.
Following the same procedure on the other side, making sure we include this negative for negative 10, negative 10 and negative six share a common factor of negative two. If we take out the negative two, negative 10𝑥 becomes five 𝑥 and negative six becomes three.
Before we do anything else, let’s check our factorisation. Is 𝑥 times five 𝑥 equal to five 𝑥 squared and is 𝑥 times three equal to three 𝑥? From there, we can ask is negative two times five 𝑥 equal to negative 10 𝑥 and is negative two times three equal to negative six.
We’re almost there; we have one final step. We have 𝑥 five 𝑥 plus three minus negative two five 𝑥 plus three. We can regroup this to say 𝑥 minus two times five 𝑥 plus three. The factorised form is 𝑥 minus two times five 𝑥 plus three.
Before we finish, I want to go back and show that it doesn’t matter if you add negative 10𝑥 plus three or three minus 10𝑥. Using the expression five 𝑥 squared minus 10𝑥 plus three minus six, we’ll find the common factors of five 𝑥 squared and negative 10𝑥. Five 𝑥 squared minus 10𝑥 share a common factor of five 𝑥.
If we take the five 𝑥 out of five 𝑥 squared, only 𝑥 remains and negative 10𝑥 divided by five 𝑥 equals negative two. Checking our work, five 𝑥 times 𝑥 equals five 𝑥 squared. Five 𝑥 times negative two equals negative 10𝑥.
Now, we consider what factor is three 𝑥 and negative six share. They share a three. Three 𝑥 divided by three equals 𝑥. And negative six divided by three equals negative two. Checking our work three times 𝑥 equals three 𝑥 and three times negative two equals negative six.
This time, we have five 𝑥, 𝑥 minus two terms, and three 𝑥 minus two terms. We add five 𝑥 plus three and multiply that by 𝑥 minus two. By switching the order of negative 10𝑥 and positive three 𝑥, the only thing that’s changed is the order of our brackets. Five 𝑥 plus three times 𝑥 minus two is the same thing as 𝑥 minus two times five 𝑥 plus three. And this is the factorised form of five 𝑥 squared minus seven 𝑥 minus six.