Solving Quadratic Equations by Factorisation

This method uses the fact that ab = 0 => a = 0 or b = 0.
Thus x² -3x +2 = 0 => (x -1)(x -2) = 0 => x -1 = 0 or x -2 = 0
=> x = 1 or x = 2. Thus x = 1, x = 2 are roots of the equation x² -3x +2 = 0.

Illustrative Examples

Example

Solve for x: (x +3)(x -3) = 40.

Solution

(x +3)(x -3) = 40 => x² -9 = 40
=> x² -49 = 0 => (x -7)(x +7) = 0
=> x -7 = 0 or x +7 = 0 => x = 7 or x = -7.
Hence the roots of the given equation are 7, -7.

Example

Solve(x -3)/(x +3) + (x +3)/(x -3) = 5/2

Solution

Given (x -3)/(x +3) + (x +3)/(x -3) = 5/2
To clear the fractions, multiply both sides by L.C.M. of fractions i.e. by 2(x +3)(x -3) to get
2(x -3)² +2(x +3)² = 5(x -3)(x +3)
=> 2(x² -6x +9) +2(x² +6x +9) = 5(x² -9)
=> 2x² -12x +18 +2x² +12x +18 -5x² +45 = 0
=> -x² +81 = 0 => x² -81 = 0 => (x -9)(x +9) = 0
=> x -9 = 0 or x +9 = 0
Hence the values x = 9, x = -9 satisfy the given equation.

Exercise

Solve the following equations by factorisation:

1. 4 x² = 3 x
2. x (2 x +1) = 6
3. 21 x² -8 x -4 = 0
4. 3 x² -5 x -12 = 0
5. (x² -5 x)/2 = 0
6. 2x²/3 -x/3 = 1
7. (x -4)² +5² = 13²
8. 3 (x -2)² = 147
9. x² -(p +q) x +p q = 0
10. x + 1/x = 41/20
11. (x +2)/(x + 3) = (2 x -3)/(3x -7)
12. 8/(x +3) - 3/(2 -x) = 2

Answers

1. 0, 3/4            2. -2, 3/2         3. 2/3, -2/7
4. 3, -4/3          5. 0, 5              6. 1, 3/2
7. -8, 16           8. -5, 9            9. p, q
10. 4/5, 5/4      11. -1, 5          12. -1/2, 5