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Quadratics Completed square
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You have already completed the bridging booklet
Reminder You have already completed the bridging booklet chapter 1 Expanding brackets chapter 4 Factorising chapter 5 Rearranging Chapter 6 Quadratics If you need support with any of the following you MUST see your teacher Factorising quadratics into brackets Solving Quadratics by factorising The basic methods for rearranging equations
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Quadratics 1a Completed square
KUS objectives BAT convert between completed square and normal form BAT rearrange and solve quadratics using completed square form Starter: Factorise and solve A x2 – 9x + 14 = 0 B x2 – 5x - 24 = 0 C 2x2 – 5x – 3 = 0 D 7x2 – 44x +12 = 0
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So completed square form is a ‘rearranged’ quadratic
Example 1 Any quadratic written in the form (x + p)2 + q Is in ‘completed square’ form e.g. (x – 4)2 – 5 x2 – 8x + 16 – 5 x2 – 8x + 11 x - 4 - 4x x2 +16 So completed square form is a ‘rearranged’ quadratic
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What completed square form shows is:
Y = x2 Y = (x – 4)2 Y = (x – 4)2 – 5 Two transformations!
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(x – 4)2 – 5 6. (x – 3)2 – 6 (x – 3)2 + 2 7. (x – 1)2 – 1 (x – 2)2 + 7
Practice 1 Try rearranging these into ‘normal’ form (x – 4)2 – 5 (x – 3)2 + 2 (x – 2)2 + 7 (x + 2)2 – 3 (x + 6)2 + 12 6. (x – 3)2 – 6 7. (x – 1)2 – 1 8. (x + 5)2 – 25 9. (x + 1)2 – 4 10. (x + 0.5)2 – 1.25 What are the minimum points of their graphs?
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x2 + 12x + 10 (x + 6)2 = x2 + 12x + 36 + 10 – 36 = - 26 (x + 6)2 - 26
WB 1a Rearranging INTO completed square form Write the (x – number)2 bit Work out what this is when multiplied out x2 + 12x + 10 (x + 6)2 = x2 + 12x + 36 Work out the number part at the end, careful ! + 10 – 36 = - 26 Put together (x + 6)2 - 26 Multiply out and check it works!
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WB 1a sketching the graph
x2 + 12x + 10 = (x + 6)2 - 26
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WB 1b Rearranging INTO completed square form
x2 + 7x + 15 Write the (x – number)2 bit Work out what this is when multiplied out (x + 3.5)2 = x2 + 7x Work out the number part at the end, careful ! + 15 – = 2.75 Put together (x + 3.5) Multiply out and check it works!
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WB 1b sketching the graph
x2 + 7x + 15 = (x + 3.5)
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x2 + 10x + 5 x2 + 4x + 3.5 x2 – 6x + 1 x2 + 4x + 2 x2 + 2x – 1
Practice 2 Try rearranging these into ‘completed square’ form x2 + 10x + 5 x2 + 4x + 3.5 x2 – 6x + 1 x2 + 4x + 2 x2 + 2x – 1 6. x2 – x – 3 7. x2 – 3x – 5 8. x2 + 3x – 3 9. x2 + 7x – 1 10. x2 + 5x + 1 What are the minimum points of their graphs? How do you check your answers?
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x2 + 12x + 10 = 0 (x + 6)2 – 26 = 0 (x + 6)2 = 26 (x + 6) = ± 26
WB 2a Solving a quadratic using completed square form x2 + 12x = 0 In completed square form (x + 6)2 – 26 = 0 Move the over (x + 6)2 = 26 Square root (two answers) (x + 6) = ± 26 Move the + 6 over The solution in exact surd form is x = ± 26 Change the surds to decimals using calculator x = Or x =
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x2 + 14x + 4 = 0 (x + 7)2 – 45 = 0 (x + 7)2 = 45 (x + 7) = ± 45
WB 2b Solving a quadratic using completed square form x2 + 14x + 4 = 0 In completed square form (x + 7)2 – 45 = 0 Move the over (x + 7)2 = 45 Square root (two answers) (x + 7) = ± 45 Move the + 6 over The solution in exact form is x = ± 3 5 Change the surds to decimals using calculator x = Or x =
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6. x2 – 4x – 3 x2 + 14x – 120 x2 – 13x – 68 7. x2 – 6x + 4 x2 – 2x - 8
Practice 3 Try rearranging these into ‘completed square’ form and then solving x2 + 14x – 120 x2 – 13x – 68 x2 – 2x - 8 x2 – 4.5x + 2 x2 + 5x – 36 6. x2 – 4x – 3 7. x2 – 6x + 4 8. x2 + 8x + 10 9. x2 + 4x – 6 10. x2 – 14x + 51 2 7 6, -20 3 5 17, -4 4, -2 -4 6 4, 0.5 -2 10 4, -9 7 2 Can you sketch their graphs Check your answers (use a calculator)
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One thing to improve is –
KUS objectives BAT convert between completed square and normal form BAT rearrange and solve quadratics using completed square form self-assess One thing learned is – One thing to improve is –
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END
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Challenge
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