### Factorising quadratics of the form ax2 + bx + c, where a is not equal to 1

ax2 + bx + c can be factorised to (mx + n)(px + q). The x2 term has the value a which is not equal to 1. The process is similar to factorising x2 + bx + c in Factorising Quadratics − 1, but this time finding the factors of both a and c and using a combination of these two factors to work out the x term.

• #### Example 1.   Factorise 2x2 + 12x + 10

• Here a = 2, b = 12 and c = 10

 i. Possible factors of a: (2, 1) ii. Possible factors of c: (5, 2) and (10, 1) iii. Work out the x term. As c is positive, select two factors of a and c that total to b.

Factor of a with first factor of c:

 iv. As c is postive the signs of the factors are the same as b:

 v. To check work in the opposite direction:

• #### Example 2.   Factorise 4x2 − 11x + 6

• Here a = 4, b = −11 and c = 6

 i. Possible factors of a: (2, 2) and (4, 1) ii. Possible factors of c: (3, 2) and (6, 1) iii. Work out the x term. As c is positive, select two factors of a and c that total to b.

First factor of a with first factor of c:

First factor of a with second factor of c:

Second factor of a with first factor of c:

 iv. As c is postive the signs of the factors are the same as b:

 v. To verify work in the opposite direction:

• #### Example 3.   Factorise 2x2 + 2x − 24

• Here a = 2, b = 2 and c = −24

 i. Possible factors of a: (2, 1) ii. Possible factors of c: (6, 4), (8, 3), (12, 2) and (24, 1) iii. Work out the x term. As c is negative, select two factors of a and c that differ by b.

Factor of a with first factor of c:

 iv. As c is negative the signs of the factors are different:

 v. To check work in the opposite direction:

• #### Example 4.   Factorise 3x2 − 4x − 20

• Here a = 3, b = −4 and c = −20

 i. Possible factors of a: (3, 1) ii. Possible factors of c: (5, 4), (10, 2) and (20, 1) iii. Work out the x term. As c is negative, select two factors of a and c that differ by b.

Factor of a with first factor of c:

Factor of a with second factor of c:

 iv. As c is negative the signs of the factors are different:

 v. To verify work in the opposite direction:

to: