Figure out the product of x + 2 and x + 3 using algebra tiles.
To find the product of x + 2 and x + 3 using algebra tiles, we represent each expression with tiles. The expression x + 2 is represented using one x-tile and two unit tiles, while x + 3 is represented using one x-tile and three unit tiles.
When arranged to form a rectangle, the inner area comprises one x²-tile, five x-tiles, and six unit tiles. The 5x is split as 2x + 3x, represented by two x-tiles on the right and three x-tiles below the x²-tile. The six unit tiles are arranged in a 2 by 3 array.
Thus, the product of (x + 2) and (x + 3) is x² + 5x + 6.
Explanation
The textbook context explains how to visualize the product of two linear expressions using algebra tiles by forming a rectangle. For (x + 3)(x + 4), the product x² + 7x + 12 is formed by splitting 7x into 3x + 4x and arranging 12 unit tiles in a 3x4 array. Applying the same logic to (x + 2)(x + 3), the product x² + 5x + 6 is formed by splitting 5x into 2x + 3x and arranging 6 unit tiles in a 2x3 array.
Solution Steps
Step 1: Represent x + 2 with one x-tile and two unit tiles, and x + 3 with one x-tile and three unit tiles.
Step 2: Arrange the tiles to form a rectangle. The inner rectangle will consist of one x²-tile, five x-tiles, and six unit tiles.
Step 3: Split the 5x term as 2x + 3x, represented by two x-tiles on the right and three x-tiles below the x²-tile, with the six unit tiles in a 2 by 3 array.
Step 4: The dimensions of the rectangle are x + 2 and x + 3, so the product is x² + 5x + 6.