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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.

Note: the inverse of factorising quadratics is Quadratic Expansion
Note: for a recap on Quadratic Expansion refer to Quadratic Expansion

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:

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

Example 1. Factorise 2x2 + 12x + 10

Example 2. Factorise 4x2 − 11x + 6

Example 3. Factorise 2x2 + 2x − 24

Example 4. Factorise 3x2 − 4x − 20

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

Example 1. Factorise 2x² + 12x + 10

Example 2. Factorise 4x² − 11x + 6

Example 3. Factorise 2x² + 2x − 24

Example 4. Factorise 3x² − 4x − 20