1. Equations Reducible To Quadratics

2. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0

3. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2

4. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4

5. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0

6. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0

7. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2

8. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2

9. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6

10. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions

11. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions
x   6

12. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions
x   6

13. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions
x   6

14. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions
x   6

15. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0 m 2  4m  3  0
 m  6  m  2   0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions
x   6

16. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0 m 2  4m  3  0
 m  6  m  2   0  m  3 m  1  0
m6 or m  2
x2  6 or x 2  2
x 6 no real solutions
x   6

17. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0 m 2  4m  3  0
 m  6  m  2   0  m  3 m  1  0
m6 or m  2 m  3 or m  1
x2  6 or x 2  2
x 6 no real solutions
x   6

18. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0 m 2  4m  3  0
 m  6  m  2   0  m  3 m  1  0
m6 or m  2 m  3 or m  1
x2  6 or x 2  2 3x  3 or 3x  1
x 6 no real solutions
x   6

19. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0 m 2  4m  3  0
 m  6  m  2   0  m  3 m  1  0
m6 or m  2 m  3 or m  1
x2  6 or x 2  2 3x  3 or 3x  1
x 6 no real solutions x 1
x   6

20. Equations Reducible To Quadratics
e.g. (i ) x 4  4 x 2  12  0 (ii ) 9 x  4  3x   3  0
let m  x 2 let m  3x
m2  x 4 m  3   3  3   9x
2 x 2 2x 2 x
m 2  4m  12  0 m 2  4m  3  0
 m  6  m  2   0  m  3 m  1  0
m6 or m  2 m  3 or m  1
x2  6 or x 2  2 3x  3 or 3x  1
x 6 no real solutions  x  1 or x  0
x   6

21. Exercise 8D; 1, 2ad, 3b, 4ab, 5ac, 6a, 8abi, 9a*