Quadratic expressions can often be factorised into two linear brackets. The method used depends on the form of the quadratic. The most common forms are \( x^2 + ax + b \) and \( ax^2 + bx + c \). Another important special case is the difference of two squares.

 

Factorising is the reverse of expanding.

 

 

Factorising Quadratics of the Form x² + ax + b

For an expression of the form \( x^2 + ax + b \), look for two numbers that:

• multiply to give \( b \)
• add to give \( a \)

 

For example:

$$
x^2 + 7x + 10
$$

 

The two numbers are \( 5 \) and \( 2 \), because:

$$
5 \times 2 = 10
$$

$$
5 + 2 = 7
$$

 

So the factorised form is:

$$
(x + 5)(x + 2)
$$

 

Another example:

$$
x^2 - x - 6
$$

 

The two numbers are \( -3 \) and \( 2 \):

$$
-3 \times 2 = -6
$$

$$
-3 + 2 = -1
$$

 

So the factorised form is:

$$
(x - 3)(x + 2)
$$

 

 

Factorising Quadratics of the Form ax² + bx + c

When the coefficient of \( x^2 \) is not 1, the method is similar but requires more care.

 

For example:

$$
2x^2 + 7x + 3
$$

 

Multiply \( a \) and \( c \):

$$
2 \times 3 = 6
$$

 

Find two numbers that multiply to \( 6 \) and add to \( 7 \). These are \( 6 \) and \( 1 \).

 

Rewrite the middle term:

$$
2x^2 + 6x + x + 3
$$

 

Factorise by grouping:

$$
2x(x + 3) + 1(x + 3)
$$

$$
= (2x + 1)(x + 3)
$$

 

Another example:

$$
3x^2 - 5x - 2
$$

 

Multiply \( a \) and \( c \):

$$
3 \times -2 = -6
$$

 

The two numbers are \( -6 \) and \( 1 \). Rewriting and factorising gives:

$$
(3x + 1)(x - 2)
$$

 

 

The Difference of Two Squares

An expression is a difference of two squares if it has the form:

$$
a^2 - b^2
$$

 

This always factorises as:

$$
(a - b)(a + b)
$$

 

For example:

$$
x^2 - 9
$$

 

This is the difference of \( x^2 \) and \( 3^2 \), so:

$$
(x - 3)(x + 3)
$$

 

Another example:

$$
4x^2 - 25
$$

 

This can be written as \( (2x)^2 - 5^2 \), so:

$$
(2x - 5)(2x + 5)
$$

 

 

Key Points to Remember

For \( x^2 + ax + b \), find two numbers that multiply to \( b \) and add to \( a \).
For \( ax^2 + bx + c \), multiply \( a \) and \( c \) first, then factorise by grouping.
The difference of two squares always factorises into two brackets.

 

Being confident with these methods is essential for solving quadratic equations and simplifying algebraic expressions.