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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=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

3+2=5

3x2=6

8+2=10

8x2=16

8-2=6

8x-2=-16

5-8=-3

5x-8=-40

-8-2=-10

-8x-2=+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, -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:

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

b2=49

4ac=4×4×1=16

2a=2×4=8

 

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

-b=-8

b2=64

4ac=4×6×-1=-25

2a=2×5=10

 

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

-b=+9

b2=81

4ac=4×4×-3=-48

2a=2×4=8