Observe the following examples

1. (3x +2) (4x +3) = 3x(4x + 3) + 2(4x + 3)
                          = 12x² + 9x + 8x + 6
                          = 12x² + 17x + 6
   The coefficient of x = 17 = 9 + 8
   9 *  8 = 72 = 12 * 6


2. (2x + 3) (4x - 5) = 2x(4x - 5) + 3(4x - 5)
                           = 8x² - 10x + 12x - 15
                           = 8x² + 2x - 15
   The coefficient of x = 2 = 12 - 10
   12 *-10 = -120 = 8?? - 15

These examples suggest the following method of factorization for a general quadratic expression.

Method of factorization of quadratic expressions

1. Multiply the coefficient of x by the constant term.
2. Resolve this product into two factors such that their sum is the coefficient of x.
3. Rewrite the x term as the sum of two terms with these coefficients.
4. Group them into two parts, each containing two terms, and factorize.

Example 1

x² - 2x - 63

Here, the coefficient of x is 1 and the constant term is -63.

So, 1 *-63 = -63

Here, -9 *  7 = -63

        -2x = -9x + 7x

x² - 2x - 63 = x² - 9x + 7x - 63

                  = x ( x - 9 ) + 7 ( x - 9 )

                  = ( x - 9 ) ( x + 7 )

Example 2

Factorize 2x² + 7x + 6

Here, 2 *  6 = 12

7 = 4 + 3; 4 *  3 = 12

Therefore, 2x² + 7x + 6 = 2x² + 4x + 3x + 6

                                   = 2x (x + 2) + 3 (x + 2)

                                   = (x + 2) (2x + 3)

Example 3

Factorize 3x² - 11x + 6

3 *  6 = 18

-11x = -9x - 2x; -9 *  -2 = 18

3x² - 11x + 6 = 3x² - 9x - 2x + 6

                    = 3x ( x - 3 ) - 2 ( x - 3 )

                    = ( x - 3 ) ( 3x - 2 )

Try these questions

I. Factorize the following

1. 2x² + 7x + 6
2. 2x² + x - 6
3. 2x² - x - 6
4. 2x² - 7x + 6
5. 3x² + 17x + 20
6. 3x² - 17x + 20
7. 3x² - 17x - 20
8. 7x² - 8x - 12
9. 6x² - 5x -14
10. 3x² - 16x + 16
11. 6 - x - 2x²
12. 6 + 7x - 3x²
13. 12 - 4x - 5x²
14. 16 + 8x - 3x²
15. 3x² + 8xy + 4y²
16. 4x² + 12xy + 5y²
17. 4x⁴ - 5x² + 1
18. 9x⁴ - 40x² + 16
19. 4x²- 25x² + 36
20. 8x⁶- 65x³+ 8

Answers to Practice Problems

1. 2x² + 7x + 6 = 2x² + 4x + 3x + 6
                       = 2x ( x + 2 ) + 3( x + 2 )
                       = ( x + 2 ) ( 2x + 3 )


2. 2x² + x - 6 = 2x² + 4x - 3x - 6
                     = 2x ( x + 2 ) - 3 ( x + 2 )
                     = ( x + 2 ) ( 2x - 3 )


3. 2x² - x - 6 = 2x² - 4x + 3x - 6
                     = 2x ( x - 2 ) + 3 ( x - 2 )
                     = ( x - 2 ) ( 2x + 3 )


4. 2x² - 7x + 6 = 2x² - 4x - 3x + 6
                       = 2x ( x - 2 ) - 3 ( x - 2 )
                       = ( x - 2 ) ( 2x - 3 )


5. 3x² + 17x + 20 = 3x² + 12x + 5x + 20
                           = 3x ( x + 4 ) + 5 ( x + 4 )
                           = ( x + 4 ) ( 3x + 5 )


6. 3x² - 17x + 20 = 3x² - 12x - 5x + 20
                           = 3x ( x - 4 ) - 5 ( x - 4 )
                           = ( x - 4 ) ( 3x - 5 )


7. 3x² - 17x - 20 = 3x² + 3x - 20x - 20
                           = 3x ( x + 1 ) - 20 ( x + 1 )
                           = ( x + 1 ) ( 3x - 20 )


8. 7x² - 8x - 12 = 7x² - 14x + 6x - 12
                         = 7x ( x - 2 ) + 6 ( x - 2 )
                         = ( x - 2 ) ( 7x + 6 )


9. 6x² - 5x -14 = 6x² - 12x + 7x - 14
                        = 6x ( x - 2 ) + 7 ( x - 2 )
                        = ( x - 2 ) ( 6x + 7 )


10. 3x² - 16x + 16 = 3x² - 12x - 4x + 16
                            = 3x ( x - 4 ) - 4 ( x - 4 )
                            = ( x - 4 ) ( 3x - 4 )


11. 6 - x - 2x² = - ( 2x² + x - 6)
                      = - [ 2x² + 4x - 3x - 6 ]
                      = - [ 2x ( x + 2 ) - 3 ( x + 2 ) ]
                      = - [ ( x + 2 ) ( 2x - 3 ) ]
                      = ( x + 2 ) ( 3 - 2x )


12. 6+ 7x - 3x² = [ 3x² - 7x - 6]
                       = [ 3x² - 9x + 2x - 6 ]
                       = [ 3x - 9x + 2x - 6 ]
                       = - [ ( x - 3 ) ( 3x + 2 ) ]
                       = ( 3 - x ) ( 3x + 2 )


13. 12 - 4x - 5x² = - [ 5x² + 4x - 12 ]
                         = - [ 5x² + 10x - 6x - 12 ]
                         = - [ 5x ( x + 2 ) - 6 ( x + 2 ) ]
                         = - [ ( x + 2 ) ( 5x - 6 ) ]
                         = ( x + 2 ) ( 6 - 5x )


14. 16 + 8x - 3x² = - [ 3x² - 8x - 16 ]
                          = - [ 3x² - 12x + 4x - 16 ]
                          = - [ 3x ( x - 4 ) + 4 ( x - 4 ) ]
                          = - [ ( x - 4 ) ( 3x + 4 ) ]
                          = ( 3x + 4 ) ( 4 - x )


15. 3x² + 8xy + 4y² = 3x² + 6xy + 2xy + 4y²
                             = 3x ( x + 2y ) + 2y ( x + 2y )
                             = ( x + 2y ) ( 3x + 2y )


16. 4x² + 12xy + 5y² = 4x² + 2xy + 10xy + 5y²
                               = 2x ( 2x + y ) + 5y ( 2x + y )
                               = ( 2x + y ) ( 2x + 5y )


17. 4x⁴ - 5x² + 1 = 4x⁴- 4x² - x² + 1
                          = 4x² ( x² - 1 ) - 1 ( x² - 1 )
                          = ( x² - 1 ) ( 4x² - 1 )
                          = ( x + 1 ) ( x - 1 ) ( 2x + 1 ) ( 2x - 1 )


18. 9x⁴ - 40x² + 16 = 9x⁴- 36x² - 4x² + 16
                             = 9x² ( x² - 4 ) - 4 ( x² - 4 )
                             = ( x² - 4 ) ( 9x² - 4 )
                             = ( x + 2 ) ( x - 2 ) ( 3x + 2 ) ( 3x - 2 )


19. 4x⁴- 25x² + 36 = 4x⁴- 16x² - 9x² + 36
                            = 4x² ( x² - 4 ) - 9 ( x² - 4 )
                            = ( x² - 4 ) ( 4x² - 9 )
                            = ( x + 2 ) ( x - 2 ) ( 2x + 3 ) ( 2x - 3 )


20. 8x⁶- 65x³+ 8 = 8x⁶- 64x³- x³+ 8
                         = 8x³( x³ - 8 ) - 1 ( x³ - 8 )
                         = ( x³- 8 ) ( 8x³- 1 )
                         = ( x - 2 ) ( x² + 2x + 4 ) ( 2x - 1 ) ( 4x² + 2x + 1 )