Identities, and why (a + b)² works

A number trick that always gives 2. Take three squares in a row, say 25, 36, 49. Add the outer two and take away twice the middle: 25 + 49 – 72 = 2. Try 49, 64, 81: 49 + 81 – 128 = 2 again. "Why always 2?" Algebra can explain it.

An identity is an area. Put lengths a and b end to end and build a square of side a + b. It splits into:

• a square a²;
• a square b²;
• two rectangles ab.

So (a + b)² = a² + 2ab + b².

"But lengths are positive. Does it work for negative numbers?" Test it: a = –2, b = –3 gives (a + b)² = 25, and a² + 2ab + b² = 4 + 12 + 9 = 25. For a = –2/3, b = 3/4 both sides give 1/144.

To be sure for every number, use the distributive law: (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b². Testing a few numbers can make us believe it; the algebra proves it.

Identity or equation?

• an identity is true for every value of its variables;
• an equation need not be: x² – 1 = 24 is true only for x = 5 or x = –5.

Is (a + b)² bigger than a² + b²? The difference is exactly 2ab. So (a + b)² is larger when ab > 0, smaller when ab < 0, and equal when a or b is 0.