Factorising Linear Equations

Factorising is the opposite of expanding brackets. We are basically trying to put an equation into brackets by finding common factors. In the example on the right we identified that 4 was a common factor of 4 and 16. As a result, we put four outside the brackets. Then we have divided 4x and 16 by 4 and put the results into the brackets. If you now try to expand the brackets, you will end up with the original equation.
Example 1 To factorise this linear equation, we found the lowest common factor which is 5 and put it outside the brackets. Then we have divided 10x and 25 by 5 and used the solutions as the values inside the brackets.
Example 2 To factorise this linear equation, we have found the lowest common factor which is 7 and placed it outside the brackets. We have then divided 14x and -28 by 7 which gave us the values inside the brackets. In this problem, we have a minus sign inside the bracket.

Factorising Quadratic Equations with Two Brackets

We recommend that you have a look at expanding double brackets before trying this lesson. In quadratic equations, you often end up with having two brackets when factorising. Factorising equations with two brackets requires a bit more thinking but is still relatively easy. The first thing you want to do is to make the x² (or 2², etc.). In the example, we have x² so we put x in each pair of bracket (this will end up as            if we tried to expand the brackets.). The next step is to find factors of 4 (1,2,4). However, we also have to make sure that we can use this factor to build the nx value. First we tried 1 and 4. This gave us 4 as 1x4 is 4 but it also gave us 5x as 1x+4x=5x. So 1 and 4 doesn’t work. Then we tried 2 and 2. This gave us 4 as 2x2 is 4 and 4x as 2x+2x=4x. Consequently, 2 and 2 are the values that come after x in the brackets. If we now expand the brackets, we would get the original equation again.
Example 1 To factorise this quadratic equation, we had to make the 2x² by putting 2x into a bracket and x in the other. We identified that to get a negative x value and a positive number we must have two negative factors. We tried all factors and ended up with -5 and -3. This works as -5 x -3 = 15 and -5x - 6x = -11x. We get -6 because we have to multiply -3 by 2x and not just by x. The equation is now factorised.
Example 2 Factorising this quadratic equation is a lot easier as we only have x² and a negative square number. We start of by butting x in both brackets again. Then we we just have to put the square root of 9 (3) negative in one bracket and positive in the other. Like this we end up with -3 x +3 = -9 and -3x + 3x = 0x. This is called the “difference of two squares”. Factorising these types of equations is very easy.
Example 3 To factorise this quadratic equation, we again started of by making x² by putting x in each pair of brackets. Due to the fact that the number (20) is negative, we have to have one positive and one negative factor. We ´discovered that 4 and -5 works as 4 x -5 = -20 and 4x - 5x = -x. The equation is now factorised.
It is very important that you always remember to make sure that you have the right numbers negative and positive. Also ensure that you always consider the number in front of the x² in certain equations where there is one.

Simultaneous and Quadratic Equations

You should try to to attempt some simultaneous equations or quadratic equations. These are quite challenging but are a major topic in secondary school algebra. You can also have a look in our library to find other topics. Please share this page if you like it or found it helpful!
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Chapter 15.0:  Learning Outcomes Students will be introduced to the concept of factorisation! Students will learn how to factorise linear equations Students will learn how to factorise quadratic equations!



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