The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.

1. Completing the Square

2. Completing the Square
e.g. (i ) x 2  6 x  7  0

3. Completing the Square
e.g. (i ) x 2  6 x  7  0
x2  6x  7 move the constant

4. Completing the Square
e.g. (i ) x 2  6 x  7  0
x2  6x  7 move the constant
x 2  6 x  32  7  32 add half the coefficient of ‘x’ squared

5. Completing the Square
e.g. (i ) x 2  6 x  7  0
x2  6x  7 move the constant
x 2  6 x  32  7  32 add half the coefficient of ‘x’ squared
x 2  6 x  9  16
 x  3  16
2
factorise to a perfect square

6. Completing the Square
e.g. (i ) x 2  6 x  7  0
x2  6x  7 move the constant
x 2  6 x  32  7  32 add half the coefficient of ‘x’ squared
x 2  6 x  9  16
 x  3  16
2
factorise to a perfect square
x  3  4

7. Completing the Square
e.g. (i ) x 2  6 x  7  0
x2  6x  7 move the constant
x 2  6 x  32  7  32 add half the coefficient of ‘x’ squared
x 2  6 x  9  16
 x  3  16
2
factorise to a perfect square
x  3  4
x  3  4
x  7 or x  1

8. (ii ) ax 2  bx  c  0

9. (ii ) ax 2  bx  c  0
b c
x2  x   0
a a

10. (ii ) ax 2  bx  c  0
b c
x2  x   0
a a
b c
x  x
2
a a

11. (ii ) ax 2  bx  c  0
b c
x2  x   0
a a
b c
x  x
2
a a
2 2
x2  x        
b b c b
   
a  2a  a  2a 

12. (ii ) ax 2  bx  c  0
b c
x2  x   0
a a
b c
x  x
2
a a
2 2
x2  x        
b b c b
   
a  2a  a  2a 
2
x b  c  b
2
 
 2a  a 4a 2
b 2  4ac

4a 2

13. (ii ) ax 2  bx  c  0
b c
x2  x   0
a a
b c
x  x
2
a a
2 2
x2  x        
b b c b
   
a  2a  a  2a 
2
x b  c  b
2
 
 2a  a 4a 2
b 2  4ac

4a 2
b b 2  4ac
x 
2a 2a

14. (ii ) ax 2  bx  c  0
b c
x2  x   0
a a
b c
x  x
2
a a
2 2
x2  x        
b b c b
   
a  2a  a  2a 
2
x b  c  b
2
 
 2a  a 4a 2
b 2  4ac

4a 2
b b 2  4ac
x 
2a 2a
b  b 2  4ac
x
2a

15. (iii ) x 2  6 x  6  0

16. (iii ) x 2  6 x  6  0
 x  3 0
2

17. (iii ) x 2  6 x  6  0
 x  3 3  0
2

18. (iii ) x 2  6 x  6  0
 x  3 3  0
2
 x  3  3  x  3  3   0

19. (iii ) x 2  6 x  6  0
 x  3 3  0
2
 x  3  3  x  3  3   0
x  3  3 or x  3  3

20. (iii ) x 2  6 x  6  0
 x  3 3  0
2
 x  3  3  x  3  3   0
x  3  3 or x  3  3
Exercise 1I; 1adh, 2ch, 3adg, 4bdfh, 5bdf, 6adg, 7bc, 8*