Factorising Quadratics
Quick summary
In mathematics, factorization or factoring is the decomposition of an expression (for example, a number, or a polynomial) into a product of other expressions (known as factors), which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5;
…and the polynomial x² − 4 factors as (x − 2)(x + 2)
Factorising
To factorise an expression such as x² + 5x + 6, you need to look for two numbers that add up to make 5 and multiply to give 6. The factor pairs of 6 are:
Create two open and close brackets as shown below;
Add the x² factors in each of the brackets as shown below.
…and lastly add the two numbers that add up to make 5 and multiply to give 6. These two numbers are 2 and 5. Therefore;
Factorising x + 7x + 12
This is a very basic quadratic expression to factorise.. First we create two sets of brackets.
x² has two factors x and x which means it factorises to;
So we can place the xs in the brackets as shown below.
There is a very simple rule you have to follow when working with the rest of the expression 7x + 8.
What two numbers multiply to give +12 but when added or subtracted give +7. You need to think about the reverse of the numbers that you put into the brackets. That reverse of factorising is expanding. Here are the pairs of numbers that multiply to give 12.
Since 3 × 4 is equal to 12 and 3 + 4 = 7 we can use 3 and 4 in the brackets.
Factorising x + 2x – 15
First draw the brackets following the same steps as before.
Now ask yourself which two numbers multiply to give -15 but when added or subtracted result in +2. Here are the numbers which multiply to give -15.
Since -3 × 5 = -15 and -3 + + 5 = 2. We can use both -3 and 5 to complete the brackets.
Difference of Two Sqaures…
This section of the article explores difference of two squares. There is a few basic rules you must know when working with difference of two sqaures.
Some quadratic expressions have only a term in x² and a number such as x² – 25.
These quadratic expressions have no x term.
Using our method to factorise quadratics means we look for two numbers that multiply to make -25 and add to make 0.
The only factor pair that will work are 5 and -5. So:
[formulae id=”9″]
Let us look at this expression like this;
Which two numbers when multiplied form -16 and when added give 0?
So the factorisation becomes;
Proof
You can prove it by expanding the brackets to get the original expression.
This is known as the difference of two squares D.O.T.S
Difference – We subtract x² and 16 so that we find the difference.
You can always tell that an expression is D.O.T.S when it contains no x term in the middle.
