Maths Year 9 (Part 3)

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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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