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© Boardworks Ltd 2005 1 of 73 Expanding and factorizing quadratic expressions 1.Expanding two brackets 2.Squaring expressions 3.The difference between two squares 4.Factorizing expressions 5.Quadratic expressions

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© Boardworks Ltd 2005 2 of 73 Expanding two brackets Look at this algebraic expression: (3 + t )(4 – 2 t ) This means (3 + t ) × (4 – 2 t ), but we do not usually write × in algebra. To expand or multiply out this expression we 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 2005 3 of 73 Expanding two brackets With practice we can expand the product of two linear expressions 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 2005 4 of 73 Matching quadratic expressions 1

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© Boardworks Ltd 2005 5 of 73 Matching quadratic expressions 2

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© Boardworks Ltd 2005 6 of 73 Squaring expressions Expand and simplify: (2 – 3 a ) 2 We can write this as, (2 – 3 a ) 2 = (2 – 3 a )(2 – 3 a ) Expanding, (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 2005 7 of 73 Squaring expressions In general, ( 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

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© Boardworks Ltd 2005 8 of 73 Squaring expressions

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© Boardworks Ltd 2005 9 of 73 The difference between two squares Expand and simplify (2 a + 7)(2 a – 7) Expanding, (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 we simplify, the two middle terms cancel out. In general, ( a + b )( a – b ) = a 2 – b 2 This is the difference between two squares.

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© Boardworks Ltd 2005 10 of 73 Matching the difference between two squares

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© Boardworks Ltd 2005 11 of 73 Factorizing expressions Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. 3 x + x 2 = x (3 + x ) 2 p + 6 p 2 – 4 p 3 = 2 p (1 + 3 p – 2 p 2 ) The highest common factor of 3 x and x 2 is x.x. (3 x + x 2 ) ÷ x =3 + x The highest common factor of 2 p, 6 p 2 and 4 p 3 is 2p.2p. (2 p + 6 p 2 – 4 p 3 ) ÷ 2 p = 1 + 3 p – 2 p 2 Factorize 3 x + x 2 Factorize 2 p + 6 p 2 – 4 p 3

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© Boardworks Ltd 2005 12 of 73 Quadratic expressions A quadratic expression is an expression in which the highest power of the variable is 2. For example, x 2 – 2, w 2 + 3 w + 1,4 – 5 g 2, t2t2 2 The general form of a quadratic expression in x is: x is a variable. a is a fixed number and is the coefficient of x 2. b is a fixed number and is the coefficient of x. c is a fixed number and is a constant term. ax 2 + bx + c (where a = 0)

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© Boardworks Ltd 2005 13 of 73 Factorizing expressions Remember: factorizing an expression is the opposite of expanding it. Expanding or multiplying out Factorizing Often: When we expand an expression we remove the brackets. ( a + 1)( a + 2) a 2 + 3 a + 2 When we factorize an expression we write it with brackets.

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© Boardworks Ltd 2005 14 of 73 Factorizing quadratic expressions Quadratic expressions of the form x 2 + bx + c can be factorized if they can be written using brackets as ( x + d )( x + e ) where d and e are integers. If we expand ( x + d )( x + e ) we have, ( x + d )( x + e ) = x 2 + dx + ex + de = x 2 + ( d + e ) x + de Comparing this to x 2 + bx + c we can see that: The sum of d and e must be equal to b, the coefficient of x. The product of d and e must be equal to c, the constant term.

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© Boardworks Ltd 2005 15 of 73 Factorizing quadratic expressions 1

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© Boardworks Ltd 2005 16 of 73 Matching quadratic expressions 1

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© Boardworks Ltd 2005 17 of 73 Factorizing quadratic expressions Quadratic expressions of the form ax 2 + bx + c can be factorized if they can be written using brackets as ( dx + e )( fx + g ) where d, e, f and g are integers. If we expand ( dx + e )( fx + g )we have, ( dx + e )( fx + g )= dfx 2 + dgx + efx + eg = dfx 2 + ( dg + ef ) x + eg Comparing this to ax 2 + bx + c we can see that we must choose d, e, f and g such that: a = df, b = ( dg + ef ) c = eg

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© Boardworks Ltd 2005 18 of 73 Factorizing quadratic expressions 2

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© Boardworks Ltd 2005 19 of 73 Matching quadratic expressions 2

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© Boardworks Ltd 2005 20 of 73 Factorizing the difference between two squares A quadratic expression in the form x 2 – a 2 is called the difference between two squares. The difference between two squares can be factorized as follows: x 2 – a 2 = ( x + a )( x – a ) For example, 9 x 2 – 16 = (3 x + 4)(3 x – 4) 25 a 2 – 1 = (5 a + 1)(5 a – 1) m 4 – 49 n 2 = ( m 2 + 7 n )( m 2 – 7 n )

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© Boardworks Ltd 2005 21 of 73 Factorizing the difference between two squares

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© Boardworks Ltd 2005 22 of 73 Matching the difference between two squares

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