1. 5.5 – Completing the Square

2. Ex. Solve x2 + 10x + 25 = 0

3. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0

4. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0

5. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0

6. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 x + 5 = 0

7. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 x + 5 = 0

8. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 x + 5 = 0
x = x = -5

9. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0
x = -5

10. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR
x = -5

11. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR
x = -5

12. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR (x + 5)2 = 0
x = -5

13. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR (x + 5)2 = 0__ √(x + 5)2 = √0
x = -5

14. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR (x + 5)2 = 0__ √(x + 5)2 = √0
x = x = 0

15. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR (x + 5)2 = 0__ √(x + 5)2 = √0
x = x = 0

16. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR (x + 5)2 = 0 __ √(x + 5)2 = √0
x = x = 0
x = -5

17. Ex. Solve x2 + 10x + 25 = 0
x2 + 10x + 25 = 0
(x )(x ) = 0
(x + 5)(x + 5) = 0
x + 5 = 0 OR (x + 5)2 = 0 __ √(x + 5)2 = √0
x = x = 0
x = -5

18. Completing The Square

19. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0

20. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable?

21. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0

22. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO

23. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0

24. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0

25. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
x2 + 8x = 18

26. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
x2 + 8x = 18 ax2 + bx = -c

27. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2

28. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2

29. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2
x2 + 8x + 16 =

30. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2
x2 + 8x + 16 =
(x + 4)(x + 4) =34

31. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2
x2 + 8x + 16 =
(x + 4)(x + 4) = 34
(x + 4)2 = 34

32. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2
x2 + 8x + 16 =
(x + 4)(x + 4) = 34
(x + 4)2 = 34
x + 4 = ±√ 34

33. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2
x2 + 8x + 16 =
(x + 4)(x + 4) = 34
(x + 4)2 = ___
x + 4 = ±√ 34 x = -4 ±√ 34

34. Completing The Square
Ex. 1 Solve x2 + 8x – 18 = 0
Factorable? (x )(x ) = 0 NO
x2 + 8x – 18 = 0
ax2 + bx = -c
x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2
x2+8x+(½8)2=18+(½8)2
x2 + 8x + 16 =
(x + 4)(x + 4) = 34
(x + 4)2 = ___
x + 4 = ±√ 34 x = -4 ±√ 34

35. Ex. 2 Solve x2 + 4x + 11 = 0

36. Ex. 2 Solve x2 + 4x + 11 = 0
Factorable? (x )(x ) = 0

37. Ex. 2 Solve x2 + 4x + 11 = 0
Factorable? (x )(x ) = 0 NO
x2 + 4x + 11 = 0

38. Ex. 2 Solve x2 + 4x + 11 = 0
Factorable? (x )(x ) = 0 NO
x2 + 4x + 11 = 0
x2 + 4x = -11

39. Ex. 2 Solve x2 + 4x + 11 = 0
Factorable? (x )(x ) = 0 NO
x2 + 4x + 11 = 0
x2 + 4x = -11 ax2+bx+(½b)2=-c+(½b)2

40. Ex. 2 Solve x2 + 4x + 11 = 0
Factorable? (x )(x ) = 0 NO
x2 + 4x + 11 = 0
x2 + 4x = -11 ax2+bx+(½b)2=-c+(½b)2
x2+4x+(½4)2=-11+(½4)2
x2 + 4x + 4 =
(x + 2)(x + 2) = -7
(x + 2)2 = __
x + 2 = ±√ -7 x = -2 ± i√ 7