# Multiplying out brackets

**Expanding expressions with brackets**

In the following algebraic expression where is the expansion and multiplication rules?

**3 y(4 – 2y) **this can also be written as

**3**

*y ×*(4 – 2*y*)Therefore the times × is not usually written in algebra. So when **expanding** or **multiplying** **out** the following expression, all the terms inside the bracket are multiplied by the term outside of it.

For example:

3y(4 – 2y) = 12y – 6y²

**Expanding brackets and simplifying**

There maybe scenarios where brackets will need to be expanded then simplified.

For example: 4*x* + 6*x*(5 – *x*)

This is expanded by:

4*x* + 6*x*(5 – *x*) = 4*x* + 30*x* – 6*x²*

Then it is simplified by

4*x* + 30*x* – 6*x² = 34x – 6x²*

Points to note:

- In the following example 4–(5
*n*– 3), the minus (-) has the role of making all the terms inside the brackets as negative first. - The same rules can be applied if there are more than one bracket in the equation

**Expanding two brackets**

When expanding two brackets, the rules in principle are the same but when tackling this scenario caution should be exercised.

For example, with the following algebraic expression:

(4 + *t*)(5 – 3*t*) = (4 + *t*) × (5 – 3*t*) but the × is not written in algebra.

To expand or multiply out this expression, every term in the first bracket should be multiplied by in the second bracket

(4 + *t*)(5 – 3*t*) = 4(5 – 3*t*) + *t*(5 – 3*t*)

=20 – 12*t* + 5*t* – 3*t²*

=20 – 17*t* – 3*t² *(this is also known as a quadratic expression)

Point to note:

- This example can be expanded and simplified in one step because in any expression in the form (
*x*+*a*)(*x*+*b*), where*a*and*b*are fixed numbers, the expanded expression will have an*x*with a coefficient of*a*+*b*and the number at the end will be*a*×*b*.

**Squaring expressions**

The following example include squaring expressions:

(2 – 4*a)² *can also be written as: (2 – 4*a) × (2 – 4a) *

= (2 – 4*a)(2 – 4a) = 2(2 – 4a) – 4a(2 – 4a) *

= 4* – 8a – 8a – 16a² *

= 4* – 16a – 16a² *

Therefore

(*a + b)² = a² + 2ab + b² *

**The difference between two squares**

The difference between two squares can be expressed as follows

(*a* + b)(*a* – b) = *a²* + b²