The Missing Number

Here is ${x}^{2}+6x$ drawn as areas. There is an x by x square and two strips, each 3 by x. Put them together and they almost make a big square. What is missing?

A 3 by 3 corner is missing. Its area is 9. So ${x}^{2}+6x+9 = {\left(x+3\right)}^{2}$. An expression like this, equal to a bracket squared, is called a perfect square.

For ${x}^{2}+bx$, each strip is half of b wide. So the missing number is half of b, squared. For ${x}^{2}-10x$, it is $\left(-5\right)^2 = 25$. This number is never negative.