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© Boardworks Ltd 20111 of 13 This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation. This icon indicates teacher’s notes in the Notes field.

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© Boardworks Ltd 20102 of 13 Expanding two brackets Look at this algebraic expression: (3 + t )(4 – 2 t ) This means (3 + t ) × (4 – 2 t ), but × is not used in algebra. To expand or multiply out this expression, multiply every term in the second bracket by every term in the first bracket. (3 + t )(4 – 2 t ) =3(4 – 2 t ) + t (4 – 2 t ) =12– 6 t + 4 t – 2 t 2 = 12 – 2 t – 2 t 2 This is a quadratic expression.

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© Boardworks Ltd 20103 of 13 Using the grid method

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© Boardworks Ltd 20104 of 13 Expanding two brackets The product of two linear expressions can be expanded in fewer steps. For example, ( x – 5)( x + 2) = x2x2 + 2 x – 5 x – 10 = x 2 – 3 x – 10 Notice that –3 is the sum of –5 and 2… …and that –10 is the product of –5 and 2.

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© Boardworks Ltd 20105 of 13 Matching quadratic expressions 1

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© Boardworks Ltd 20106 of 13 Matching quadratic expressions 2

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© Boardworks Ltd 20107 of 13 Squaring expressions Expand and simplify: (2 – 3 a ) 2 (2 – 3 a ) 2 = (2 – 3 a )(2 – 3 a ) = 2(2 – 3 a ) – 3 a (2 – 3 a ) =4– 6 a + 9 a 2 = 4 – 12 a + 9 a 2

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© Boardworks Ltd 20108 of 13 Squaring expressions ( a + b ) 2 = a 2 + 2 ab + b 2 The first term squared… …plus 2 × the product of the two terms… …plus the second term squared. For example, (3 m + 2 n ) 2 = 9 m 2 + 12 mn + 4 n 2 It can be seen that there is a pattern relating a squared expression and its form once expanded and simplified.

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© Boardworks Ltd 20109 of 13 Squaring expressions

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© Boardworks Ltd 201010 of 13 The difference between two squares Expand and simplify: (2 a + 7)(2 a – 7) (2 a + 7)(2 a – 7) = 2 a (2 a – 7) + 7(2 a – 7) = 4a24a2 – 14 a + 14 a – 49 = 4 a 2 – 49 When simplifying, the two middle terms cancel out. ( a + b )( a – b ) = a 2 – b 2 This is the difference between two squares.

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© Boardworks Ltd 201011 of 13 The difference between two squares

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© Boardworks Ltd 201012 of 13 The difference between two squares

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© Boardworks Ltd 201013 of 13 Pendants of gold Here are two pendant patterns which are to be made out of gold. The white squares are holes. 2x2x 5 x 1 x + 1 2x2x 3 x + 2 2 x + 2 1 x For what value of x do they both use the same amount of gold? Do they always use the same amount of gold?

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