Quadratic Equations

• An expression of the form ax² +bx +c, a0 is called a quadratic or second degree expression in x.
• A quadratic equation has two and only two roots, which may or may not be different.
• Some quadratic equations can be solved by factorisation.
• Formula for solving quadratic equation ax² +bx +c = 0 is
       x = [-b ± (b²-4ac)]/2a
• = b² -4ac is called discriminant.

Exercise

1. Which of the following are quadratic equations:
   (i) 2x² +3x +4 = 0
   (ii) -3x² = 5x +1
   (iii) x² +3 = x² +3x
   (iv) x -1/x = 2
   (v) (x -1)(x -2) = 2
2. Determine whether x = 2 is a solution of the following equations:
   (i) x² -4x +4 = 0
   (ii) (x -2)(x -3) = 0
   (iii) (x -2)(x -3) = 2
   (iv) 2 x² = 2 2 x
   (v) x² +x +1 = 0
3. Find the values of a and b such that x = 2, x = -1 are solutions of the quadratic  x² +ax +b = 0.

Solve the following quadratic equations (4 -15) by factorisation:

4. (x +3)(x -3) = 40
5. x (2x +5) = 25
6. 4x² = 3x
7. x²-5x = 0
8. 3x² -5x -12 = 0
9. 3x² = x +4
10. 2x² -x = 3
11. (x -4)² +5² = 13²
12. [(3x -5) -5)²]/7 = 28
13. 4 3 x² +5x -23 = 0
14. x² -(1 +2) x +2 = 0
15. 3x -8/x = 2

Solve for x the following equations (16 -25):

16. a/(x -b) + b/(x - a) = 2
17. a/(ax -1) + b/(bx -1) = a + b, where a + b 0, ab 0
18. 1/p + 1/q + 1/x = 1/(p +q +x), where p + q 0, p 0, q 0
19. x/(x-1) + (x -1)/x = 5/2
20. 8/(x +3) - 3/(2 - x) = 2
21. (x +2)/(x +3) = (2 x -3)/(3 x -7)
22. [x (x -7)] = 32
23. [x +15] = x +3
24. [3x² -2 x -1] = [2x -2]
25. Find the values of x if p -2 = 0, q +15 = 0 and x² +px +q = 2

Solve the following quadratics by using formula (26 -33):

26. x² -6x +9 = 0
27. 2x² +7x +6 = 0
28. (2x +3)(3x -2) +2 = 0
29. 25x² +30x +7 = 0
30. (x -2)/(x +2) + (x +2)/(x - 2) = 4
31. (x -1)/(x -2) + (x -3)/(x -4) = 10/3
32. 2/(x+2) - 1/(x+1) = 4/(x +4) - 3/(x +3)
33. a(x² +1) = (a² +1) x, a 0

Answers