The Missing Number

Often you have only the first two terms, like x^2 + 10x, and want to make a perfect square. The pattern x^2 + 2px + p^2 says what is missing: the middle number is 2p, so p is half of it, and the last number must be p squared.

So for x^2 + bx, the missing number is (b/2)^2. For x^2 + 10x, half of 10 is 5 and 5 squared is 25, so x^2 + 10x + 25 = (x + 5)^2. For x^2 - 12x, half of -12 is -6, and -6 squared is 36: the number you add is positive even when b is negative.

An odd b works the same way, with a fraction. For x^2 + 5x, half of 5 is 5/2 and its square is 25/4, so x^2 + 5x + 25/4 = (x + 5/2)^2. This missing number is the heart of the next lesson.