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
- 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:
- 4x² = 3x
- x²-5 x = 0
- x (2x +1) = 6
- x² -3x -10 = 0
- 3x² -5x -12 = 0
- x² -2x +1 = 0
- 3x² = x +4
- 2 x² -x = 3
- 6 +x -x² = 0
- 4x² +4x +1 = 0
- (x -3)(2x +5) = 0
- (x +2)(x -3) = 6
- x +1/x = 41/20
- 3x -8/x = 2
- (x +2)/(x +3) = (2 x -3)/(3 x - 7)
- 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