Expanding brackets

Expanding basic single brackets

To expand a simple bracket multiply the numbers in the bracket by the number outside the bracket. The basic formula is:

a (b + c) = ab + a c

For example:

3 (x + 2) = 3 x + 3 \cdot 2 = 3 x + 6

Here are some other examples:

2 (4 x + 5) 5 (2 x + 3) = 2 \cdot 4 x + 2 \cdot 5 = 8 x + 10 = 5 \cdot 2 x + 5 \cdot 3 = 10 x + 15

Expanding double brackets

To expand double brackets, its the same process as before but with two brackets. The formula is:

(a + b) (c + d) = a c + a d + b c + b d (x + y) (x + z) = x^{2} + x z + x y + yz

For example:

(2 x + 8) (x + 4) = 2 x \cdot x + 2 x \cdot 4 + 8 \cdot x + 8 \cdot 4 = 2 x^{2} + 8 x + 8 x + 32

Grid method to expanding brackets

Some people find it hard to expand brackets using the formula above, so they use the grid method. This method involves drawing a grid and filling in the products of the terms from each bracket.

(2 x + 8) (x + 4) 2 x^{2} + 8 x + 8 x + 32 = \times 2 x 8 x 2 x^{2} 8 x 4 8 x 32 = 2 x^{2} + 16 x + 32

Then you add up all the terms in the grid:

Expanding more complex double brackets

When you have more complex double brackets, you can still use the same formula. For example:

(a + b) (c + d + e) = a c + a d + a e + b c + b d + b e

We can simplify using Collecting like terms to make the result more manageable. For example:

(2 x + 3) (x + 4 + 5) = 2 x \cdot x + 2 x \cdot 4 + 2 x \cdot 5 + 3 \cdot x + 3 \cdot 4 + 3 \cdot 5 = 2 x^{2} + 8 x + 10 x + 3 x + 12 + 15 = 2 x^{2} + (8 x + 10 x + 3 x) + (12 + 15) = 2 x^{2} + 21 x + 27