Factorise quadratics

We have looked at factorising quadratic where the coeffect of the \( x^2 \) term is 1. If we have \( 5x^2 \), we will need to follow a slightly different method to factorise.

1. Write the quadratic in standard form:

Ensure the quadratic is written as \( ax^2 + bx + c \).

2. Identify \( a \), \( b \), and \( c \):

Recognise the values of \( a \) (the coefficient of \( x^2 \)), \( b \) (the coefficient of \( x \)), and \( c \) (the constant term).

3. Find two numbers that multiply to \( ac \) and add to \( b \):

Multiply \( a \) and \( c \). Find two numbers that multiply to this \( ac \) value and add up to \( b \). These two numbers will help split the middle term.

4. Rewrite the quadratic by splitting the middle term:

Replace \( bx \) with two terms using the numbers found in the previous step. This splits the quadratic into four terms. For example, if the quadratic is \( 6x^2 + 7x + 2 \), find two numbers that multiply to \( 6 \times 2 = 12 \) and add to \( 7 \), which are \( 3 \) and \( 4 \). Rewrite \( 6x^2 + 7x + 2 \) as \( 6x^2 + 3x + 4x + 2 \).

5. Factorise by grouping:

Group the terms into two pairs: \( (6x^2 + 3x) + (4x + 2) \). Factor out the common factor from each pair: \( 3x(2x + 1) + 2(2x + 1) \).

6. Factorise out the common binomial:

If both groups have a common factor (a binomial), factor it out. For the example, factor out \( (2x + 1) \): \( (2x + 1)(3x + 2) \).

7. Check the factorisation:

Expand the factors to confirm they multiply to the original quadratic.

Example

Factorise \( 6x^2 + 7x + 2 \).

1. Write in standard form:

\( 6x^2 + 7x + 2 \).

2. Identify \( a = 6 \), \( b = 7 \), \( c = 2 \):

The values are \( a = 6 \), \( b = 7 \), and \( c = 2 \).

3. Find two numbers that multiply to \( 6 \times 2 = 12 \) and add to \( 7 \):

The numbers \( 3 \) and \( 4 \) work.

4. Rewrite the expression:

Rewrite \( 6x^2 + 7x + 2 \) as \( 6x^2 + 3x + 4x + 2 \).

5. Group terms:

\( (6x^2 + 3x) + (4x + 2) \).

6. Factor out the common factor:

Factor out \( 3x(2x + 1) + 2(2x + 1) \).

7. Factor out \( (2x + 1) \):

\( (2x + 1)(3x + 2) \).

The factorised form is:

\( (2x + 1)(3x + 2) \).