# Maths Year 9 (Part 3)

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- Created by: carmen.lau
- Created on: 11-06-15 21:32

## Area & Surface area

Similar Shapes:

- Scale factor for area: ( S.f. for length)
- S.f for volume: ( S.f. for volume)

- Eg. To find a:
- Length S.f. = 18 -- 12 = 1.5
- a = 135 -- (1.5)
- a = 60cm

- Eg. To find b:
- Length S.f. = 10 -- 5 = 2
- Volume S.f. = 2 = 8
- b = 7 x 8 = 56

- Eg. To find c:
- Volume S.f. = 300 -- 24 = 125
- Length S.f. = 125 = 5

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## Factorising Quadratic Equations

**Using this method, you can factorise expressions in the form:**

**ax + bx + c**

- Eg. 6x + x - 12 = 0 ( Instructions in red)
- Times 'a' and 'c': 6 x -12 = -72
- Find two numbers that multiply to make the -72 and add to make 1: -8 , 9
- Create a new equation, copying 'a' and 'c' from the original: 6x - 8x + 9x - 12 = 0
- Factorise: 2x ( 3x - 4 ) + 3 ( 3x - 4 )
- Both equations in the brackets should be the same and create a new equation:
- ( 3x - 4 ) ( Same as the equation in the brackets) ( 2x + 3 ) ( Other two numbers ):
- ( 3x -4 ) ( 2x + 3 )

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## Completing the Square

**Using this method, you can factorise equations in the form:**

**ax + bx + c**- With 'c' being a prime number
- 'bx' is called the
**co-efficient** - 'c' is called the
**constant**

- Eg. x + 6x + 3 (Instructions in red)
- Factorise 'ax + bx' : (x + 3)
- Halve the co-efficient and add( subtract if -ve) to the factorised equation: (x + 3) + 3
- Subract the square of the number in the brackets: (x + 3) + 3 - 9
- Simplify: (x + 3) - 6

**Solving the equation:**Eg. x + 6x - 13 = 0- Follow the steps above to complete the square: (x + 3) - 22 = 0
- Rearrange the equation- first add 22: (x + 3) = 22
- Square root (Must add a ' ' to the beginning of the surd): x + 3 = 22
- Minus 3 & simplify surd if possible. Leave answer in the form: a b: x = -3 22

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## Completing the Square (Part 2)

**Completing the square with fractions:**Eg. x + x - 5 = 0- Complete the square in the same way: (x + 1/2) - 1/4 - 5 (This is also 20/4) = 0
- Simplify: (x + 1/2) - 21/4 = 0
- Rearrange: (x + 1/2) = 21/4
- x + 1/2 = 21/4
- Simplify: x + 1/2 = 21/ 4 = 21/2
- Rearrange: x = -1/2 21/2
- Simplify: x = (-1 21) / 2

**Completing the square with multiple x's:**Eg. 2x + 10x + 6- Divide everything by 2: 2 (x + 5x + 3)
- Complete the square in the same way: 2 [ (x + 5/2) - 25/4 + 3]
- Simplify: 2 [ (x + 5/2) - 13/4]
- Times by 2 ( to get rid of brackets): 2 (x + 5/2) - 13/2 (The last fraction's denominator is halved)

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## Completing the Square (Part 3)

**Negative x :**We need a positive x to complete the square:- Eg. - x + 4x + 6
- Factorise using '- 1' (changes every sign to the opposite): - ( x - 4x - 6)
- Complete the square in the same way: - [ (x - 2) - 4 - 6 ]
- Simplify: - [ (x - 2) - 10 ]
- Times by -1 to get rid of the bracket: (x - 2) + 10 (Sign before '10' changes)

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## Quadratic Formula

**In general, a quadratic equation takes the form:**

**ax + bx + c = 0**

**You use the formula:**

**x = - b b - 4ac MUST LEARN!**

- Eg. 5x - 11x - 4 = 0
- Find a, b & c: a = 5 , b = - 11 , c = - 4
- Rewrite the formula with a, b & c: x = - ( - 11) ( - 11) - 4 ( 5) ( - 4)

- Simplify: x = 11 21 + 80 ( Work out the square root first)

- Work out: x = 11 201 (The ' ' creates 2 answers)

- x =11 201 OR 11 201

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## Direct Proportion

- Both increase or decrease at the same time
- We use a multiplier, called 'k'
- The sign is used to show proportionality:
**C U**

**To make an equation:**Eg. C = 60 , U = 300- C = kU
- Rearrange ( Make 'k' the subject): k = C / U
- Sub in: k = 60 / 300
- Work out: k = 1/5
- Rearrange for the equation: C = U / 5
**To work out an answer:**If U = 235:- C = 235 / 5 ( 5 is the multiplier)
- C = 47

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## Direct Proportion: Squares

**C r **

**To work out an equation with x :**C = 68 , r = 2- C = kr
- Rearrange: k = c / r
- Sub in & Work out: k = 68 / 2
- k = 68 / 4 = 17
- Rearrange: C = 17r
- Use this equation to work out answers

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## Inverse Proportion

- When one increases, the other decreases and visa versa
- This is shown by:
**y 1 / x**(The fraction is always: 1 / [ A letter] )

**To create an equation:**Eg. L 1 / W L = 30 , W = 20- Rearrange the equation: L = k ( multiplier = L x W ) / W
- L = ( 30 x 20) / W
- L = 600 / W

**Squares**: Eg. Y = k / X Y = 3 , X = 2- Rearrange the equation: Y = k ( Mulltiplier = Y x X ) / X
- Y = ( 3 x 2 ) / 2
- Y = 12 / 2 = 12 / 4
- Y = 3

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