The Lesson

Expanding double brackets means removing brackets from a pair of expressions in brackets. Imagine we wanted to remove the brackets from the expression below. We have to expand the brackets. ( x plus 1 ) ( x plus 2 ) We can use the grid method to expand the brackets.

Expanding Double Brackets Using the Grid Method

Look at the grid below. It is a rectangle. It represents (x + 1)(x + 2). grid representing ( x plus 1 ) ( x plus 2 ) Consider the entire rectangle. entire rectangle
  • The height of the entire rectangle is x + 1.
  • The width of the entire rectangle is x + 2.
  • The area of the rectangle is the height × width. It is (x + 1)(x + 2).
Now consider the four smaller rectangles that make up the entire rectangle. Consider the dark blue rectangle. dark rectangle
  • The height of the dark blue rectangle is x.
  • The width of the dark blue rectangle is x.
  • The area of the dark blue rectangle is x2.
Consider the light blue rectangle. light rectangle
  • The height of the light blue rectangle is x.
  • The width of the light blue rectangle is 2.
  • The area of the light blue rectangle is 2x.
Consider the orange rectangle. orange
  • The height of the orange rectangle is 1.
  • The width of the orange rectangle is x.
  • The area of the orange rectangle is x.
Consider the pink rectangle. pink
  • The height of the pink rectangle is 1.
  • The width of the pink rectangle is 2.
  • The area of the pink rectangle is 2.
The area of the dark blue, light blue, orange and pink rectangles must add up to the area of the entire rectangle.
(x + 1)(x + 2) = x2 + 2x + x + 2

A Real Example of How to Expand Double Brackets Using the Grid Method

Expanding brackets is easy. Expanding double brackets using the grid method is easy.

Question

Expand the brackets below.
x ( y plus 2 )

Step-by-Step:

1

Find the first term in the first brackets (x). Draw a vertical line and label its height with this term. vertical line labelled x

2

Find the second term in the first brackets (2). Draw a vertical line down from the vertical line already drawn. Label its height with this term. vertical line labelled 2

3

Find the first term inside the second brackets (y). Draw a horizontal line from the vertical lines already drawn. Label its width with this term. horizontal line labelled y

4

Find the second term inside the second brackets (3). Draw another horizontal line, following the horizontal line already drawn. Label its width with this term. horizontal line labelled 3

5

Draw the rest of the grid. grid

6

Find the area of each rectangle in the grid by multiplying the height and width of each rectangle. The area of the first grid is its height (x) times its width (y).
x × y = xy
x times y equals xy

7

Find the area of each rectangle in the grid by multiplying the height and width of each rectangle. The area of the second grid is its height (x) times its width (3).
x × 3 = 3x
x times 3 equals 3x

8

Find the area of each rectangle in the grid by multiplying the height and width of each rectangle. The area of the third grid is its height (2) times its width (y).
2 × y = 2y
2 times y equals 2y

9

Find the area of each rectangle in the grid by multiplying the height and width of each rectangle. The area of the fourth grid is its height (2) times its width (3).
2 × 3 = 6
2 times 3 equals 6

9

Add the areas of the rectangles in the grid.
xy + 3x + 2y + 6

Answer:

Expanding (x + 2)(y + 3) gives xy + 3x + 2y + 6.

Lesson Slides

The slider below shows another real example of how to expand double brackets using the grid method.

Multiplication Tips

  • Multiplying a number with a variable

    2 × x = 2x

    5 × x = 5x

  • Multiplying a number with a variable with a coefficient

    2 × 2x = 4x

    3 × 5x = 15x

  • Multiplying a variable with a variable

    x × y = xy

    x × x = x2