Special Products a(x + y + z) = ax + ay + az (x + y)(x – y) = x 2 – y 2 (x + y) 2 = x 2 + 2xy +y 2 (x – y) 2 = x 2 – 2xy +y 2 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x – y) 3 = x 3 – 3x 2 y + 3xy 2 – y 3

Solving Linear Equations

Isolating the Variable on the Right 2x - 4 = 4x - 10

Isolating the Variable on the Right 2x – 4 = 4x – 10 2x – = 4x 2x + 6 = 4x 6 = 2x (½)6 = (½)2x 3 = x

CHECKING 2x – 4 = 4x – 10Solution x = 3 2(3) – 4 = 4(3) – 10? 6 – 4 = 12 – 10OK

Parentheses 3 – ( r – 1 ) = 2 (r + 1 ) – r

Parentheses 3 – ( r – 1 ) = 2 (r + 1 ) – r 3 – r + 1 = 2r + 2 – r 4 – r = r – r + r = r r 4 = 2r = 2r 1 = r

Parentheses Check 3 – ( r – 1 ) = 2 (r + 1 ) – r 1 = r 3 – ( [1] – 1 ) = 2 ([1] + 1 ) – [1]? 3 = 2(2) – 1? 3 = 4 – 1OK

Strategy for Solving Equations

- x = c – x = c=> x = – c – x = 2=> x = – 2

Algebra - substitution or evaluation Given an algebraic equation, you can substitute real values for the representative values Perimeter of a rectangle is P = 2L + 2W If L = 3 and W = 5 then: P = 2   5 = = 16

Substitution A joiner earns £W for working H hours Her boss uses the formula W = 5H + 35 to calculate her wage. Find her wage if she works for 40 hours W = 5  = = £235

Substitution Find the value of 4y - 1 when y = 1/4 0 y = Find F = 5(v + 6) when v = 9 75

Rearranging formulae Sometimes it is easier to use a formula if you rearrange it first y = 2x + 8 Make x the subject of the formula Subtract 8 from both sides y  8 = 2x Divide both sides by 2 ´2y - 4 = x

Rearranging formulae A = 3r 2 Make r the subject of the formula Divide both sides by 3 A/3 = r 2 Take the square root of both sides  A/3 = r

Brackets The milkman’s order is 3 loaves of bread, 4 pints of milk and 1 doz. Eggs per week Suppose the cost of bread is b, the cost of milk is m and a dozen eggs is e. Work out the cost after 5 weeks = 5(3b + 4m + e)

Brackets = 5(3b + 4m + e) to ‘remove’ brackets, each term must be multiplied by 5 5(3b + 4m + e) = 15b + 20m + 5e If the number outside the bracket is a negative, take care: the rules for multiplication of directed numbers must be applied

Brackets 4(3x - 2y) 4  3x - 4  2y = 12x - 8y What about -2(x-3y)? = -2  x - (-2)  3y = -2x + 6y

Factorising The opposite of multiplying out brackets Need to find the common factors Very important - it enables you to simplify expressions and hence make it easier to solve them A factor is a number which will divide exactly into a given number.

Factorising 2x + 6y 2 is a factor of each term (part) of the expression and therefore of the whole number. 2x + 6y = 2  x + 2  3  y = 2  (x +3  y) = 2(x + 3y)

Factorising 6p + 3q + 9r 3 is the common factor = 3(2p + q + 3r) x 2 + xy + 6x x is the common factor for each term x(x + y + 6)