1. For Educational Use Only © 2010 12.4 Completing the Square Brian Preston Algebra 1 2009-2010

2. For Educational Use Only © 2010 ? Real World Application How far away from the end of the board can you dive into the water? 10 ft

3. For Educational Use Only © 2010 Lesson Objectives 1) Solve a quadratic equation by completing the square. 2) Choose a method for solving a quadratic equation.

4. For Educational Use Only © 2010 Example 1) What are the ways to solve a quadratic equation (ax 2 + bx + c = 0)? Square Roots Graphing Factoring Quadratic Formula Completing the Square

5. For Educational Use Only © 2010 Definition Completing the Square x2x2 + bx+ = x + b 2 b 2 ) ( 2 ) ( 2 To make this more manipulative we will add a term. Factoring the left terms will get you the right terms.

6. For Educational Use Only © 2010 Definition Completing the Square x2x2 + bx+ = x + b 2 b 2 ) ( 2 ) ( 2 To make this more manipulative we will add a term. Perfect Square There has to be a 1 to do this process. 1

7. For Educational Use Only © 2010 (x + 3) 2 Definition Completing the Square using algebra tiles. x 2 + 6x Make a square x2x2 x xx x x x x 1 1 1 1 1 1 1 1 1 x 2 + 6x + 9 =+ ( ) 2 6 2 6 x + 3 x x

8. For Educational Use Only © 2010 Example Find the term that should be added to the expression to create a perfect square trinomial. 2) x2x2 – 8x+ – 8 2 ) ( 2 or 16 1

9. For Educational Use Only © 2010 Example 3) x2x2 + 0.4x+ 0.4 2 ) ( 2 or 0.04 Find the term that should be added to the expression to create a perfect square trinomial. 1

10. For Educational Use Only © 2010 – 0.44x 2 + 2.61x + 10 4) The path of a diver diving from a 10-foot high diving board is h = – 0.44x 2 + 2.61x + 10 where h is the height of the diver above water (in feet) & x is the horizontal distance (in feet) from the end of the board. How far away from the end of the board will the diver enter the water? h = – 0.44x 2 + 2.61x + 10 far 10 h = – 0.44x 2 + 2.61x + 10 0 = Real World Application h far

11. For Educational Use Only © 2010 2.61 –0.44 2.61 –0.44 b (2.61) b a 10 2 – 4 (–0.44) (10) Solve – 0.44x 2 + 2.61x + 10 = 0 ax 2 + bx + c = 0 a = b = 2.61 – 0.44 c = 10 x = b 2 – 4ac – b ± 2a x = – ± 2 (2.61) c a Standard form Real World Application 4)

12. For Educational Use Only © 2010 4) x = 6.8121 x = + 17.6 ± – 0.88 – 2.61 24.4… x =x = ± – 0.88 –2.61 4.94… ± – 0.88 –2.61 = Real World Application (2.61) 2 – 4 (–0.44) (10) – ± 2 (2.61)

13. For Educational Use Only © 2010 x = 4.94… ± – 0.88 – 2.61 4.94… + – 0.88 – 2.61 = 4.94… – – 0.88 – 2.61 = – 2.65 8.58 x = 8.58 feet 4) Real World Application

14. For Educational Use Only © 2010 Real World Application How far away from the end of the board can you dive into the water? 10 ft 8.58ft ?

15. For Educational Use Only © 2010 Example 5) x2x2 + 10x= Solve the equation by completing the square. 24 1

16. For Educational Use Only © 2010 + ( ) 2 10 2 (x + 5) 2 Definition Completing the Square using algebra tiles. x 2 + 10x Make a square x2x2 x xx x x x x 1 1 1 1 1 1 1 1 1 x 2 + 10x + 25 =10 x + 5 x x x x x x x 1 1 1 x 1 1 1 1 1 1 1 1 1 1 1 1 1

17. For Educational Use Only © 2010 Example 5) x2x2 + 10x= 10 2 ) ( 2 Solve the equation by completing the square. + 24 10 2 ) ( 2 + x2x2 + 10x= + 24 +5252 5252 x + ) ( 2 5 = 49 5 x + 5 = 7 ± 10

18. For Educational Use Only © 2010 – 5 Example 5) Solve the equation by completing the square. x + 5 = 7 ± x – 5 = 7 ± x = + 7 x – 5 = – 7 = 2 = – 12 x = 2,– 12

19. For Educational Use Only © 2010 – 4 Example 6) x2x2 – 6x+ Solve the equation by completing the square. 4 = 20 – 4 x2x2 – 6x = 16 1

20. For Educational Use Only © 2010 Definition Completing the Square using algebra tiles. x 2 – 6x Make a square x2x2 x xx x x x x 1 1 1 1 1 1 1 1 1 x 2 – 6x + 9 + ( ) 2 – 6 2 x x

21. For Educational Use Only © 2010 (x – 3) 2 Definition Completing the Square using algebra tiles. x 2 – 6x Make a square x2x2 x xx x x x 1 1 1 1 1 1 1 1 1 x 2 – 6x + 9 = x – 3

22. For Educational Use Only © 2010 –3 – Example 6) x2x2 – 6x= -6 2 ) ( 2 Solve the equation by completing the square. + 16 -6 2 ) ( 2 + x2x2 6x= + 16 + –3 x – ) ( 2 3 = 25 x – 3 = 5 ± – 6 )2)2 ()2)2 (

23. For Educational Use Only © 2010 + 3 Example 6) Solve the equation by completing the square. x – 3 = 5 ± x 3 = 5 ± x 3 = + 5 x 3 = – 5 = 8 = – 2 x = 8,– 2

24. For Educational Use Only © 2010 + 3 Example 7) x2x2 – x– Solve the equation by completing the square. 3 = 0 + 3 x2x2 – x = 3 1

25. For Educational Use Only © 2010 1 2 2 2 – Example 7) x2x2 – x= 2 ) ( 2 Solve the equation by completing the square. + 3 2 ) ( 2 + x2x2 1x= + 3 + )2)2 x – ) ( 2 = x – = ± 1 1 2 ()2)2 ( 2 – 1 13 4 2

26. For Educational Use Only © 2010 x – = ± 1 2 13 2 + + + Example 7) Solve the equation by completing the square. x = ± 1 2 1 2 1 2 13 2 1 2

27. For Educational Use Only © 2010 x – = ± 1 2 13 2 ++ Example 7) Solve the equation by completing the square. x = ± 1 2 1 2 1 2 13 2

28. For Educational Use Only © 2010 2 22 + 13 Example 8) 2x 2 – 8x– Solve the equation by completing the square. 13 = 7 + 13 2x 2 – 8x = 20 1 22 x2x2 – 4x = 10

29. For Educational Use Only © 2010 + ( ) 2 Definition Completing the Square using algebra tiles. x 2 – 4x Make a square x2x2 x x x x x 1 1 1 1 x 2 – 4x + 4 – 4 2 x

30. For Educational Use Only © 2010 (x – 2) 2 Definition Completing the Square using algebra tiles. x 2 – 4x Make a square x2x2 x x x x 1 1 1 1 x 2 – 4x + 4 = x – 2

31. For Educational Use Only © 2010 4x –2 – Example 8) x2x2 – = -4 2 ) ( 2 Solve the equation by completing the square. + 10 -4 2 ) ( 2 + x2x2 4x= + 10 + –2 x – ) ( 2 2 = 14 x – 2 = ± )2)2 ()2)2 ( – 4

32. For Educational Use Only © 2010 + 2 Example 8) Solve the equation by completing the square. x – 2 = ± x 2 = ± 14

33. For Educational Use Only © 2010 + 2 Example 8) Solve the equation by completing the square. x – 2 = ± x 2 = ± 14

34. For Educational Use Only © 2010 4 44 + 11 Example 9) 4x 2 + 4x– Solve the equation by completing the square. 11 = 0 + 11 4x 2 + 4x = 11 1 44 x2x2 + 1x = 11 4

35. For Educational Use Only © 2010 11 4 1 2 1 2 1 2 1 2 + Example 9) x2x2 + x= 1 2 ) ( 2 Solve the equation by completing the square. + 1 2 ) ( 2 + x2x2 1x= + + )2)2 x + ) ( 2 = x + = 1 1 2 ()2)2 ( 1 11 4 3 3 ±

36. For Educational Use Only © 2010 –– – x + = 1 2 Example 9) Solve the equation by completing the square. x = 1 2 1 2 1 2 1 2 3 ± 3 ± –

37. For Educational Use Only © 2010 x + = 1 2 –– Example 9) Solve the equation by completing the square. x = 1 2 1 2 1 2 3 ± 3 ± –

38. For Educational Use Only © 2010 3 3 + 15 10) 3x 2 – 15 = 0 Choose a method to solve the quadratic equation. Example + 15 3x 2 = 15 3 x2x2 = 5 x = 5 ±

39. For Educational Use Only © 2010 11) x 2 + 12x + 20 = 0 1 Factors of 20 1  20 + 12 = 2  10 Choose a method to solve. Standard form 1 + 20 21 = 20 – 1 19 = Example 4  5

40. For Educational Use Only © 2010 8 11) x 2 + 12x + 20 = 0 1 2 + 10 +12 = Choose a method to solve. 1  20 Factors of 20 2 + 10 12 = 10 – 2 = Example 2  10 4  5

41. For Educational Use Only © 2010 1x +10 + 2 11) x 2 + 12x + 20 = 0 1 +2 2 + 10 10 2 ( +12 = Choose a method to solve. 1  20 Factors of 20 )( ) 1x 2 = 0 Example 2  10 4  5

42. For Educational Use Only © 2010 – 10 (x + 10) (x + 2) (x + 10) (x + 2) – 2 x + 10 Choose a method to solve. 11) = 0 x + 2 x – 10 = ( ) ( ) x – 2 = Example

43. For Educational Use Only © 2010 – 26 2 – 4 2 22 (1) (– 26) a (2) 12) Solve x 2 + 2x – 26 = 0 1 ax 2 + bx + c = 0 a = b = 2 1 1 c = – 26 x = b 2 – 4ac – b ± 2a x = – ± 2 (2) bb c a 11 Standard form Example

44. For Educational Use Only © 2010 (1) (– 26) (2) 12) x = 2 – 4 – ± 2 (2) 4 x = + 104 ± 2 – 2 108 x = ± 2 – 2 10.39 ± 2 – 2 = Example

45. For Educational Use Only © 2010 x = 10.39 ± 2 – 2 10.39 + 2 – 2 = 10.39 – 2 – 2 = 8.39 2 = 4.196 – 12.39 2 = – 6.196 x = 4.196 & – 6.196 12) Example

46. For Educational Use Only © 2010 Key Points & Don’t Forget 1) Don’t forget the negative signs. 2) Add to both sides of the equation 3) When taking the square root of an equation you will get a ± answer. b 2 ) ( 2

47. For Educational Use Only © 2010 The Assignment pg. 624-626 #’s 8-58 even, 62-65

48. For Educational Use Only © 2010 Bibliography Please email brianspowerpoints@gmail.com with errors, confusing slides, improvements, complications, or any other comments or questions.brianspowerpoints@gmail.com http://www.worldofteaching.comhttp://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching.