Chapter 11.5 — Exercise 11.4 — Algebraic Identities

Four Identities, Derived Once and Reused Forever

Multiplying out (a + b) × (a + b) the long way, exactly as Exercise 11.3 would, gives a² + ab + ab + b², which combines to a² + 2ab + b². Since this holds for every value of a and b, it earns identity status:

The same expansion with a minus sign throughout gives a second identity:

Multiplying a sum by a difference is different from either of those — the two middle terms (−ab and +ab) cancel completely instead of combining:

A fourth identity handles two binomials that share the same variable but different constants:

Every one of these four is just an ordinary binomial-times-binomial expansion, done once in general terms instead of with specific numbers — the entire value of an identity is that this expansion never needs repeating once its pattern is recognised in a new problem.

Matching a Product to the Right Identity

Picking which of the four identities applies is mostly a matter of reading the signs and structure of what's being multiplied. Eight examples cover every pattern:

The last row shows the fourth identity applied more loosely than a first glance suggests: treating 2b as the shared "x," with −a and +c playing the role of the two constants, (2b−a)(2b+c) matches the (x+a)(x+b) pattern even though the "constants" here are themselves letters rather than plain numbers.

Turning Big Multiplications Into Small Ones

The real payoff of these identities shows up in arithmetic, not algebra — numbers close to a round hundred or thousand square almost instantly once they're seen as (round number ± small number):

Squaring 304 by hand the ordinary way means multiplying two three-digit numbers directly; rewriting it as (300+4)² turns the same problem into one squared round number, one small multiplication, and one small square — all easy enough to do mentally, with no long multiplication at all.

Products That Aren't Squares at All

The a² − b² identity turns certain products into subtraction instead of squaring, which is useful whenever two numbers sit equally spaced above and below a convenient midpoint:

Both products look like ordinary two-digit or three-digit multiplication at first glance, but recognising that 304 and 296 sit exactly 4 above and below 300 (and that 83 and 77 sit exactly 3 above and below 80) turns each into a single squaring and a single subtraction.

Two Numbers Near the Same Base, Not Equally Spaced

When two numbers sit close to the same round base but aren't symmetric around it, the fourth identity — (x+a)(x+b) — takes over from a² − b²:

Here 9 and 8 (or 4 and 6) aren't opposite in sign the way 304/296 or 83/77 were — both numbers sit above the same base of 100 (or 200) instead of one above and one below — which is exactly the signal to reach for (x+a)(x+b) instead of a² − b².

Telling the Four Patterns Apart at a Glance

With eight worked examples now behind it, a short rule of thumb sums up how to choose among the four identities without re-deriving any of them: two identical brackets with a plus sign inside call for (a+b)²; two identical brackets with a minus sign call for (a−b)²; one bracket with + and a matching one with − call for a²−b²; and two brackets sharing one term but differing constants — same base, different amounts added — call for (x+a)(x+b). Every one of the eight numeric evaluations above is really just this same lookup, applied to numbers dressed up to look like ordinary multiplication.

Why an Identity Doesn't Need Checking Every Time

It's worth being clear about what makes these four results different from an ordinary calculation. Once (a+b)² ≡ a²+2ab+b² has been proven true for every possible a and b — not just checked for a few sample numbers — it can be applied to a brand-new pair of values, like 300 and 4, with total confidence and no need to re-verify it against that specific pair first. This is the real distinction between an identity and a plain equation: a² − 4 = 0 only holds for two particular values of a, so it says nothing at all about what happens at a = 3 or a = 10, but (a+b)² ≡ a²+2ab+b² holds at every single value simultaneously, which is exactly why it was safe to reach for it directly when squaring 304 or 509 without first testing it on those specific numbers.

From an Algebraic Shortcut to a Geometric One

Every identity here was derived by direct algebraic multiplication — the (a+b)² identity, for instance, is exactly the binomial expansion from Exercise 11.3 generalised into a reusable rule. Exercise 11.5 shows that the same four identities can be proven a completely different way — by cutting up and rearranging areas of squares and rectangles — which is often what makes an identity feel obviously true rather than just algebraically true — seeing why a square's area splits into exactly those four pieces tends to stick far longer than the multiplied-out algebra alone.