The Machine in Letters

The rule for factoring can be written with letters. The identity x^2 + (p + q)x + pq = (x + p)(x + q) holds for any numbers p and q. The coefficient of x is their sum, and the constant term is their product.

Expanding (x + p)(x + q) gives x^2 + (p + q)x + pq. Factoring uses the same identity in the other direction. It turns the sum back into the product.

Expand (x + p)(x + q) to get x^2 + px + qx + pq. Combine the two x terms to get x^2 + (p + q)x + pq. This shows why the two numbers must add to the middle coefficient and multiply to the constant.