The Power of Expanding Brackets

Expanding brackets is a crucial algebra skill that helps you simplify expressions. Think of it like unpacking a gift: everything inside the brackets needs to be multiplied by the value outside. Let’s explore how to do this step by step, whether you're working with single or double brackets.

 

Expanding Single Brackets

When you expand a single bracket, you multiply every term inside the bracket by the term outside. This is often called the distributive law.

General Rule

For $a(b + c)$: $$a(b + c) = ab + ac$$

 

Example 

Expand: $3(x + 4)$

Step-by-Step Solution:

1. Multiply $3$ by $x$: $3 \times x = 3x$
2. Multiply $3$ by $4$: $3 \times 4 = 12$
3. Write the result: $$3(x + 4) = 3x + 12$$

 

 

 

 

Expanding Double Brackets

When expanding double brackets, every term in the first bracket is multiplied by every term in the second bracket. This is called the FOIL method (First, Outside, Inside, Last).

General Rule

For $(a + b)(c + d)$:

$$(a + b)(c + d) = ac + ad + bc + bd$$

 

 

Example

Expand: $(x + 3)(x + 5)$

Step-by-Step Solution:

1. Multiply the First terms: $x \times x = x^2$
2. Multiply the Outside terms: $x \times 5 = 5x$
3. Multiply the Inside terms: $3 \times x = 3x$
4. Multiply the Last terms: $3 \times 5 = 15$
5. Add them all together: $$(x + 3)(x + 5) = x^2 + 5x + 3x + 15$$
6. Simplify the middle terms $(5x + 3x = 8x)$: $$(x + 3)(x + 5) =  x^2 + 8x + 15$$

 

Example

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

Step-by-Step Solution:

1. Multiply the First terms: $2x \times x = 2x^2$
2. Multiply the Outside terms: $2x \times 4 = 8x$
3. Multiply the Inside terms: $-1 \times x = -x$
4. Multiply the Last terms: $-1 \times 4 = -4$
5. Add them all together: $$(2x - 1)(x + 4) = 2x^2 + 8x - x - 4$$
6. Simplify the middle terms $(8x - x = 7x)$: $$(2x - 1)(x + 4) = 2x^2 + 7x - 4$$

 

 

 

 

Expanding Special Cases

Perfect Squares

For $(a + b)^2$: $$(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2$$

Example: $$(x + 3)^2 = x^2 + 6x + 9$$

 

Difference of Two Squares

For $(a + b)(a - b)$: $$(a + b)(a - b) = a^2 - b^2$$

Example: $$(x + 5)(x - 5) = x^2 - 25$$