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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
Contents
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=x^{2} 

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

3+2=5 3x2=6 
_{} 
8+2=10 8x2=16 

82=6 8x2=16 

58=3 5x8=40 

82=10 8x2=+16 
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, 1x2, 2x1 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: 1x5, 1x5, 5x1 or 5x1. 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, 1x3, 3x1 
Solving linear equations:

Add 1 to both sides (6+1=7) Divide by 5 

Subtract 4x ;
(8x4x=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 x^{2} positive
Where you can factorise:
or 
Factorise Solve each bracket equal to zero 
or 
Important: Subtract 2 to make the equation equal to zero. Then factorise and solve each bracket equal to zero 
or 
Factorise 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.
QUADRATIC FORMULA:
To solve:
Use

a=4, b=7, c=1 b=7 b^{2}=49 4ac=4×4×1=16 2a=2×4=8 

a=5, b=8, c=1 b=8 b^{2}=64 4ac=4×6×1=25 2a=2×5=10 

a=4, b=9, c=3 b=+9 b^{2}=81 4ac=4×4×3=48 2a=2×4=8 