Quadratics Completed Square Methods
Explore the concept of completing the square in quadratics, understand rearranging equations, and practice converting between completed square and normal form. Get a grasp of solving quadratics by factorising and delve into identifying minimum points on graphs. Ideal for mastering quadratic equations.
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Presentation Transcript
Quadratics Completed square
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
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 x x2 - 4x +16 - 4 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!
Practice 1 Try rearranging these into normal form 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 1. (x 4)2 5 2. (x 3)2 + 2 3. (x 2)2 + 7 4. (x + 2)2 3 5. (x + 6)2 + 12 What are the minimum points of their graphs?
WB 1a Rearranging INTO completed square form x2 + 12x + 10 Write the (x number)2 bit Work out what this is when multiplied out (x + 6)2 = x2 + 12x + 36 Work out the number part at the end, careful ! + 10 36 = - 26 (x + 6)2 - 26 Put together 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 (x + 3.5)2 + 2.75 Put together Multiply out and check it works!
WB 1b sketching the graph x2 + 7x + 15 = (x + 3.5)2 + 2.75
Practice 2 Try rearranging these into completed square form 1. x2 + 10x + 5 2. x2 + 4x + 3.5 3. x2 6x + 1 4. x2 + 4x + 2 5. 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?
WB 2a Solving a quadratic using completed square form x2 + 12x + 10 = 0 (x + 6)2 26 = 0 In completed square form Move the - 26 over (x + 6)2 = 26 (x + 6) = 26 Square root (two answers) Move the + 6 over The solution in exact surd form is x = - 6 26 x = - 0.901 Or x = - 11.1 Change the surds to decimals using calculator
WB 2b Solving a quadratic using completed square form x2 + 14x + 4 = 0 (x + 7)2 45 = 0 In completed square form Move the - 26 over (x + 7)2 = 45 (x + 7) = 45 Square root (two answers) Move the + 6 over The solution in exact form is x = - 7 3 5 x = Change the surds to decimals using calculator Or x =
Practice 3 Try rearranging these into completed square form and then solving 2 7 6. x2 4x 3 x2 + 14x 120 1. 6, -20 3 5 x2 13x 68 7. x2 6x + 4 17, -4 2. 4, -2 x2 2x - 8 8. x2 + 8x + 10 3. -4 6 4, 0.5 x2 4.5x + 2 4. 9. x2 + 4x 6 -2 10 4, -9 x2 + 5x 36 5. 7 2 10. x2 14x + 51 Can you sketch their graphs Check your answers (use a calculator)
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
