Expanding two linear expressions means multiplying two brackets, each containing linear terms. This can involve one variable or two different variables. The aim is to remove the brackets and simplify the result.

 

The method used is to multiply every term in the first bracket by every term in the second bracket.

 

 
Expanding Two Brackets with One Variable

Consider the expression:

$$
(x + 3)(x + 5)
$$

 

Multiply each term in the first bracket by each term in the second bracket:

$$
x \times x + x \times 5 + 3 \times x + 3 \times 5
$$

 

Simplify each term:

$$
x^2 + 5x + 3x + 15
$$

 

Collect like terms:

$$
x^2 + 8x + 15
$$

 

Another example:

$$
(2x - 1)(x + 4)
$$

 

Multiply each term:

$$
2x \times x + 2x \times 4 - 1 \times x - 1 \times 4
$$

 

Simplify:

$$
2x^2 + 8x - x - 4
$$

$$
= 2x^2 + 7x - 4
$$

 

 
Expanding Two Brackets with Two Variables

Expressions may also contain more than one variable.

 

For example:

$$
(x + y)(x + 2y)
$$

 

Multiply each term:

$$
x \times x + x \times 2y + y \times x + y \times 2y
$$

 

Simplify:

$$
x^2 + 2xy + xy + 2y^2
$$

 

Collect like terms:

$$
x^2 + 3xy + 2y^2
$$

 

Another example:

$$
(2a + b)(a - 3b)
$$

 

Multiply each term:

$$
2a \times a - 2a \times 3b + b \times a - b \times 3b
$$

 

Simplify:

$$
2a^2 - 6ab + ab - 3b^2
$$

$$
= 2a^2 - 5ab - 3b^2
$$

 

 
Key Points to Remember

Multiply every term in the first bracket by every term in the second bracket.
Be careful with signs, especially when subtracting.
Collect like terms at the end to simplify fully.

 

Expanding two linear expressions correctly is essential for simplifying algebraic expressions and solving equations later on.