Quadratic Equations

• An expression of the form ax² +bx +c where a 0 is called a quadratic (or second degree) expression in variable x. An equation of the type ax² +bx +c = 0, a 0, is called a quadratic equation in variable x.
   Thus x² -1 = 0; 1 -2x +3x² = 0 are quadratic equations.
• A number is called a root or solution of quadratic equation ax² +bx +c = 0 if it satisfies given equation, that is if a² +b +c = 0.

Solving quadratic equations by factorisation

• This method involves using a property of numbers known as zero product rule:
       If ab = 0 then a = 0 or b = 0.
  Hence we may factorise a quadratic expression and use above rule to solve the quadratic equation.

Exercise

1. Determine whether x = 1/2 and x = 3/2 are the solutions of the quadratic 2x² -5x +3 = 0 or not.

     Solve the following quadratic equations by factorisation:

2. 4x² = 3x
3. x²-5 x   = 0
4. x (2x +1) = 6
5. x² -3x -10 = 0
6. 3x² -5x -12 = 0
7. x² -2x +1 = 0
8. 3x² = x +4
9. 2 x² -x = 3
10. 6 +x -x² = 0
11. 4x² +4x +1 = 0
12. (x -3)(2x +5) = 0
13. (x +2)(x -3) = 6
14. x +1/x = 41/20
15. 3x -8/x = 2
16. (x +2)/(x +3) = (2 x -3)/(3 x - 7)
17. 8/(x +3) - 3/(2 -x) = 2

Answers

1. x = 1/2 is not a solution; x = 3/2 is a solution.
2. 0, 3/4               3. 0, 5           4. -2, 3/2
5. -2, 5                6. 3, -4/3        7. 1, 1
8. -1, 4/3             9. -1, 3/2       10. -2, 3
11. -1/2, -1/2     12. 3, -5/2      13. -3, 4
14. 4/5, 5/4         15. 2, - 4/3 .   16. -1, 5
17. -1/2, 5