Now consider the case where we have a rectangle of sidelengths 2x + 3 and 3x + 1, as shown in the figure. What can you say about its area (2x + 3) (3x + 1)?

Step-by-step solution

Idea: The area of the rectangle equals the total area of the tiles inside it. Count each kind of tile, then confirm with the distributive property.

1. Big squares: the 2x part of the width and the 3x part of the height make a 2 × 3 block of x²-tiles: 6x².½ mark
2. x-tiles: the 3 units of width beside the 3x height give 3 × 3 = 9 x-tiles; the 1 unit of height under the 2x width gives 2 x-tiles. Total 11x.½ mark
3. Unit tiles: the corner is 3 × 1, so 3 unit tiles.½ mark
4. Area = 6x² + 11x + 3. Check: (2x + 3)(3x + 1) = 6x² + 2x + 9x + 3 = 6x² + 11x + 3 ✓.½ mark

Check: Put x = 1: (5)(4) = 20 and 6 + 11 + 3 = 20 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing the x² term as 5x² (adding 2 and 3). The coefficients multiply: 2 × 3 = 6.
• Counting only 9 x-tiles and missing the 2 along the bottom (or the other way round).
• Writing the constant as 3 + 1 = 4. The corner is 3 × 1 = 3.

How this can come in the exam

Try one yourself

Find the area of a rectangle with sides (4x + 1) and (x + 3).

More questions like this

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