Quadratics Completed square

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

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

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

What completed square form shows is: Y = x2 Y = (x – 4)2 Y = (x – 4)2 – 5 Two transformations!

(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?

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!

WB 1a sketching the graph x2 + 12x + 10 = (x + 6)2 - 26

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 + 12.25 Work out the number part at the end, careful ! + 15 – 12.25 = 2.75 Put together (x + 3.5)2 + 2.75 Multiply out and check it works!

WB 1b sketching the graph x2 + 7x + 15 = (x + 3.5)2 + 2.75

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?

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 + 10 = 0 In completed square form (x + 6)2 – 26 = 0 Move the - 26 over (x + 6)2 = 26 Square root (two answers) (x + 6) = ±  26 Move the + 6 over The solution in exact surd form is x = - 6 ±  26 Change the surds to decimals using calculator x = - 0.901 Or x = - 11.1

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 - 26 over (x + 7)2 = 45 Square root (two answers) (x + 7) = ±  45 Move the + 6 over The solution in exact form is x = - 7 ± 3 5 Change the surds to decimals using calculator x = Or x =

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)

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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