Algebra tiles and (x + a)(x + b)

A rectangle with sides x + 3 and x + 4 has area (x + 3)(x + 4) = x² + 3x + 4x + 12 = x² + 7x + 12.

Algebra tiles make it visible. There are three kinds of tile:

• a big x² square;
• x-tiles, strips x long and 1 wide;
• unit tiles, 1 by 1.

Put 3 x-tiles along one side of the x² tile and 4 along the other. The corner fills with a 3 by 4 block of 12 unit tiles.

So the tiles work both ways: multiplying (x + 3)(x + 4), and factoring x² + 7x + 12 back into (x + 3)(x + 4).

The split of the middle term matters. Splitting 7x as 2x + 5x would need a 2 by 5 corner of 10 unit tiles, not 12, so the tiles do not make a rectangle. The two parts must add to 7 and multiply to 12.

The pattern: (x + a)(x + b) = x² + (a + b)x + ab. Check: (x + 6)(x + 7) = x² + 13x + 42.

With numbers in front: (px + a)(qx + b) = pqx² + (pb + aq)x + ab. For (2x + 3)(3x + 1): 6x² + (2 + 9)x + 3 = 6x² + 11x + 3.