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Core 1: Algebraic Skills Part 1

Key facts

·        Every student must realise that mastering key algebraic skills is essential to success in Mathematics.

·        Many of the later skills and topics require a solid understanding of the topics covered in this section.

·        It is therefore advisable to study the following topics carefully and in detail and in the order presented.

·        Practising the examination style questions will be of extreme benefit as the examinations approach



1)           Collecting like terms

2)          Expanding brackets

3)          Factorising (easier)

4)          Factorising (Harder)

5)          Solving linear equations

6)          Solving quadratic equations


In each section, there are two columns to the tables. Column one shows the algebra, column two shows the thought process and things to consider.




Collecting like terms



Just one letter


Combine the coefficients


Work out the bracket




Still just one letter


Write down the correct sign


Work out the bracket



Two letters!


Combine for x and y separately


Work out both brackets




Three letters (or more...)


Look at each separately


Work out all brackets




Different looking terms



Observe the same process


All you have to do is add or subtract


Then work it all out. Easy!


Expanding brackets

The 5 means multiply both terms in the bracket by 5


Multiply both terms by 7

Remember that 7 multiplied by -2 is -14


Multiply both terms by 8



Multiply the coefficients (numbers together)

Remember that x×x=x2



Multiply the second bracket by BOTH TERMS in the first bracket separately!


Multiply the second bracket by BOTH TERMS in the first Bracket.

Remember it is -3 multiplied by 4 and -3 multiplied by +2.

This is why we get -12 and -6




Linear factorising


10 is the HCF (highest common factor) of 10 and 40

5 is the HCF of 15 and 35

Observe/remember to keep the minus sign


3 is the HCF of 3 and 12


is another factor


6 is HCF of 12 and 18


 is another factor













Quadratic factorising: Easier













Harder factorising: There are algorithms for doing these however if you can build up a natural instinct for how these work then it is better for you in the long term.

After a while, your instinct starts to give you the correct answer more quickly

Options are


Begin to expand:

This one is the correct option so no need to try others

To get the 3 we must have


To get the final 2 could have:

2x1, 1x2, -1x-2, -2x-1


Expand each option until you find the correct one

Try options

; incorrect

; incorrect

; incorrect

; correct at last!

To get the 3 we must have



To get -5 could have:

1x-5, -1x5, -5x1 or 5x-1.


Try each until you find the correct option

Try some options:

; incorrect

; correct!


To get the 7 we must have


To get +3 could have

1x3, 3x1, -1x-3, -3x-1

Solving linear equations:


Add 1 to both sides (6+1=7)

Divide by 5


Subtract 4x ;    (8x-4x=4x)

Add 3 ;              (-9+3=-6)

Divide by 4 and simplify



Expand first

Collect like terms

Subtract 11x

Subtract 8


Find the LCM (Lowest common multiple) of 5, 2, 6 and convert all fractions to have this denominator. LCM=30

Then multiply by the LCM, it just cancels out.

Then solve as normal


Insert brackets on any numerator with more than one term and imagine all terms as fractions.

Find the LCM of the denominators, in this case 10.

Multiply by the LCM and solve as normal

Solving quadratic equations:

Remember: always rearrange to make the equation equal to zero

Where possible, make the coefficient of x2 positive

Where you can factorise:




Solve each bracket equal to zero



Important: Subtract 2 to make the equation equal to zero.

Then factorise and solve each bracket equal to zero




Be careful when solving equations when the coefficient of x doesn’t equal 1


Factorising is generally the easiest and quickest way to solve quadratic equations.

There is another way to solve an equation of this type. This works on all quadratic equations whether you can factorise it or not. This is by using the QUADRATIC FORMULA.



To solve:




a=4, b=7, c=1






a=5, b=8, c=-1






a=4, b=-9, c=-3