Now let us try to factorise 2 6 4 x xy 29 29 2 2 4 6

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after expansion. Now let us try to factorise 2 6 4 x xy + . { } { } { } { } ( 29 ( 29 2 2 4 6 6 4 6 4 6 4 3 2 2 2 2 3 2 2 3 2 × × + = × + × = × × × + × × × = × × + × = + E5555F E5555F &&& &&& E555555555F E555555555F xy x x x x y x xy x x x y x x x y x x y x x y . You don’t have to go through these steps to get your answer. This is just to show you how we arrived to our solution. This question could have been answered in one go, as bellow.
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16 ( 29 2 6 4 2 3 2 x xy x x y + = + . Now let us try to factorise (a) 2 2 x y xy - and (b) 3 2 7 x x + . (a) { } { } ( 29 ( 29 2 2 2 2 - - = × × + × - × = - = - &&& &&& &&& E5555555F E55555555F x y xy x y xy x x y x y y x y x y xy x y . Or in one go: ( 29 2 2 x y xy xy x y - = - . (b) { } { } ( 29 ( 29 3 2 3 2 2 7 7 7 7 7 + = × × + × × = + = + &&& &&& &&& E5555555F E5555555F x x x x x x x x x x x x x x Or in one go ( 29 3 2 2 7 7 x x x x + = + Extra Exercises for Practice Factorise 2 2 3 2 2 2 2 2 ( ) 2 ( ) ( ) 1 ( ) 2 18 ( ) 2 9 - + - - + - - a a ab b b x x c z d p p e q q Exercise 1.3.1 Factorise ( ) 5 25 a x - ; ( ) 27 18 b x + ; ( ) 8 24 c x - + ; 2 ( ) 4 d x x + ; 2 2 ( ) e x y xy + ; 2 ( ) 4 d x x + ; 2 2 ( ) e x y xy + ; 2 ( ) 4 4 f a x a - ; 2 ( ) 24 8 g a x a +
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17 2. Solving Equations We will try to solve three types of equations: linear, quadratic, and simultaneous. For this section we might need to use some of the tools we have learned in the previous sections. 2.1 Solving Linear Equations A linear equation is one that can be written in the form 0 ax b + = where a and b are numbers and the unknown quantity is x. For example 4 1 0 x + = and 1 3 0 2 x - = are both linear equations. Linear equations may also appear in the following forms (which appeared to be different from 0 ax b + = but are equivalent): 2 1 3 x - = ; 3 4 1 x x - + = + ; and ( 29 3 2 6 0 x - + = . These equations can be rearranged to get the form 0 ax b + = , as shown bellow. 2 1 3 x - = 2 1 3 3 3 x - - = - 2 4 0 x - = Now let us move to the second expression: 3 4 1 x x - + = + 3 4 1 x x - + = + Subtract 3 from both sides so that the right hand side becomes 0 Simplify this expression This expression is of the form 0 ax b + = Simplify this expression by bearing in mind that when you move on the other side of the “=” you change sign change
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18 4 1 3 5 2 x x x - - = - ⇒ - = - 5 2 2 2 x - + = - + 5 2 0 x - + = Finally, ( 29 3 2 6 0 x - + = can also be expressed in the form 0 ax b + = . ( 29 3 2 6 0 x - + = 6 18 0 x - + = Now let us try to solve some linear equations: (a) 10 0 x + = ; (b) 5 100 0 - + = x ; (c) 4 6 3 x - = - ; (d) ( 29 3 2 5 x + = ; and (e) 5 2 2 3 x x + = - - . (a) 10 0 10 x x + = = - (as you can see if you replace x by –10 you get 0) (b) 100 5 100 0 5 100 20 5 x x x - - + = ⇒ - = - = = - (c) 3 4 6 3 4 3 6 4 3 4 x x x x - = - ⇒ = - + = = (d) ( 29 1 3 2 5 3 6 5 3 5 6 3 1 3 x x x x x - + = + = = - = - ⇒ = (e) 5 5 2 2 3 5 2 3 2 7 5 7 x x x x x x - + = - - + = - - = - ⇒ = . Add +2 to both sides so that the right hand side is 0 Simplify This expression is of the form 0 ax b + = Expand the brackets This expression is of the form 0 ax b + =
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