Special Binomial Products

Some people can say $35^2 = 1225$ instantly. Their secret is not lightning mental arithmetic but a handful of memorized special products: split 35 into $30 + 5$ and a formula finishes the job in three steps. Today we welcome the two most useful ones home.

The classic trap

Try a quick experiment: are $(3+4)^2$ and $3^2 + 4^2$ the same? The first is $7^2 = 49$ , the second only $9 + 16 = 25$ — a gap of 24. Distributing a square onto each term inside a bracket is the most stubborn misconception left over from expanding brackets.

Expand honestly, everything times everything:

$(a+b)^2 = (a+b)(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$

That middle term $2ab$ is exactly the source of the gap between 25 and 49 ( $2 \times 3 \times 4 = 24$ ). Swap $b$ for $-b$ and you get the second identity:

$(a-b)^2 = a^2 - 2ab + b^2$

For instance $(x-3)^2 = x^2 - 6x + 9$ — the middle term carries a minus sign, and it is still not allowed to vanish.

See 2ab in a picture

No need to memorize blindly. Draw a big square of side $(a+b)$ and slice it once horizontally and once vertically into four pieces: two squares $a^2$ and $b^2$ , plus two identical rectangles $ab$ . The total area $a^2 + ab + ab + b^2$ is exactly the right-hand side of the formula.

Try numbers: $a = 3$ , $b = 4$ gives pieces of 9, 12, 12 and 16, totalling $49 = 7^2$ . The two 12s are $2ab$ . On the grid below, set both width and height to 7 and count out 49 unit squares — then picture the two cuts. This jigsaw works for any $a$ and $b$ .

InteractiveArea Grid

Width w5Height h4

Area = length × width=5 × 4 = 20unit squares

Difference of squares, by cutting and pasting

The third identity is even slicker: $(a+b)(a-b)$ . Expanding gives $a^2 + ab - ab - b^2$ , and the middle terms cancel, leaving

$(a+b)(a-b) = a^2 - b^2$

Seen as area: cut a corner of side $b$ off an $a \times a$ square and an L-shaped piece of area $a^2 - b^2$ remains; slide the vertical strip over next to the horizontal one and it becomes an exact $(a+b) \times (a-b)$ rectangle — cut one piece, patch another, area unchanged.

It is also a mental-math superpower: $42 \times 38 = (40+2)(40-2) = 1600 - 4 = 1596$ .

Use the formulas, then check them

Expanding (2x+3)²

Treat $2x$ as $a$ and 3 as $b$ : $(2x)^2 + 2 \cdot 2x \cdot 3 + 3^2 = 4x^2 + 12x + 9$ . Check with $x = 1$ : the left side is $(2+3)^2 = 25$ and the right side is $4 + 12 + 9 = 25$ — they agree. The checking craft lives in substitution, the firmest insurance a formula user can buy.

Where did the middle term go

Two high-frequency errors: $(a+b)^2 \neq a^2 + b^2$ and $(a-b)^2 \neq a^2 - b^2$ — both correct answers carry a middle term $\pm 2ab$ . Before expanding, ask one question: where is the middle term?

The formulas also run in reverse: meet $x^2 + 6x + 9$ , recognize $(x+3)^2$ — and that is already the job of factorising.

Check yourself

Quick quiz

1. What is (x + 4)² expanded?
2. What is (x + 7)(x − 7) expanded?
3. Use the difference of squares to compute 43 × 37 in your head. What is the result?