Maths Year 9 (Part 3)

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

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 )

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

• 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)

• 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)

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

• 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

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

• 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