Every teacher in every country has at some point watched a student confidently write (a + b)^2 = a^2 + b^2 on the board. It is the most common algebra mistake in school arithmetic, and it happens because the rule looks like it should follow the distributive pattern a(b + c) = ab + ac. It doesn't. Squaring a sum is a completely different operation, and the missing middle term — the one that sneaks out when you skip a step — is almost always the part of the problem that matters.

The correct identity, one line

(a + b)^2 = a^2 + 2ab + b^2.

Three terms, not two. The outer a^2 and b^2 are the "obvious" pieces. The 2ab in the middle is the part people drop. Once you see why it is there, you will never drop it again.

Where the middle term comes from

(a + b)^2 is shorthand for (a + b)(a + b). Two copies of the bracketed sum, multiplied together. Expand it honestly, one distribution at a time:

(a + b)(a + b) \;=\; a(a + b) + b(a + b).

Why: the distributive rule says (x)(y + z) = xy + xz. Here the x is the whole first bracket (a + b), and the second bracket's two terms a and b are the things it distributes across. So the first bracket multiplies a, then multiplies b, and the two results are added.

Now expand each piece separately using distribution once more:

a(a + b) = a^2 + ab \qquad b(a + b) = ab + b^2.

Add them:

a^2 + ab + ab + b^2 \;=\; a^2 + 2ab + b^2.

The two ab terms are where the middle 2ab comes from. It is not a bonus — it is what happens when you multiply a by b once coming from the left bracket and again coming from the right bracket. You genuinely multiply a and b together twice, and the two copies add.

The picture — a square cut into four rectangles

The identity is not just algebra. It is a statement about area. Draw a square of side a + b. Its area is (a + b)^2 — one number, the total. Now cut the square with one horizontal line at height a and one vertical line at width a. You have four pieces: an a \times a square in one corner, a b \times b square in the opposite corner, and two a \times b rectangles filling the other two corners.

A square of side a plus b cut into four pieces showing the identityA big square whose side length is labelled a plus b. One horizontal and one vertical cut divide it into four coloured pieces: an a-by-a square in the top-left, an a-by-b rectangle in the top-right, a b-by-a rectangle in the bottom-left, and a b-by-b square in the bottom-right. The pieces are labelled a squared, a b, a b, and b squared. The total area is written below: a squared plus two a b plus b squared. a² a · b a · b b² a b a b total area = a² + 2ab + b²
A square of side $a + b$ cut into four pieces. The two corner squares have areas $a^2$ and $b^2$. The two rectangles each have area $a \times b$. Add the four pieces and you get $(a + b)^2 = a^2 + 2ab + b^2$. The two rectangles are the $2ab$ term — two of them, each of area $ab$, and they cannot disappear without cutting away real area.

If you tried to claim (a + b)^2 = a^2 + b^2, you would be saying the big square is made up of only the two corner squares — and you would be throwing away two entire rectangles of area ab each. The picture makes it impossible to cheat.

Drag it and watch the gap

Interactive gap between a squared plus b squared and a plus b all squaredA horizontal number line from zero to ten with two draggable red points labelled a and b. Four live readouts above the line display the values of a, b, a squared plus b squared (the wrong answer), and a plus b all squared (the correct answer), along with the gap between them equal to two a b. As the reader drags the points, the gap grows and shrinks, always staying exactly equal to two a b and never becoming zero unless a or b is zero. 0 5 10 ↔ drag either point
Drag $a$ and $b$ between $0$ and $10$. The readout shows $a^2 + b^2$ (the wrong answer) next to $(a + b)^2$ (the right one) and the gap between them. That gap is always exactly $2ab$ — and it only vanishes when one of $a$ or $b$ is zero. The two are not approximately equal. They are not equal "most of the time." They are equal only when one of the inputs is zero, and in every other case the middle term $2ab$ is the whole story.

The sanity check — use numbers before you trust the algebra

The single best habit for catching this mistake is a five-second numerical check. Any time you are tempted to write (a + b)^2 = a^2 + b^2, pause and plug in two small numbers.

Try a = 2, b = 3.

(2 + 3)^2 = 5^2 = 25 \qquad 2^2 + 3^2 = 4 + 9 = 13.

25 \neq 13. Off by 12 = 2 \cdot 2 \cdot 3 = 2ab. Exactly the middle term you would have dropped. This check takes four seconds and makes the error impossible to miss.

Why this works: any claimed identity \text{LHS} = \text{RHS} must agree for every pair of values. One disagreeing pair disproves the claim. The swap-test idea (see the swap test) generalises: concrete numbers kill wrong algebra identities in a single line.

Why the mistake feels natural

The trap is that students generalise the pattern of the distributive law — which is a real, working rule — past the place it applies.

a(b + c) = ab + ac \quad \text{(this works)}

Multiplication distributes over addition when one factor is sitting outside the bracket and being multiplied into a sum. But squaring is not the same as distributing. When you write (a + b)^2, the entire bracket (a + b) is being multiplied by another copy of itself, not by a single number. So the two-way cross-multiplication kicks in and you get the extra 2ab.

The deeper rule: exponents do not distribute over addition. (a + b)^n \neq a^n + b^n for any n > 1 (unless a or b is zero). For n = 2 the discrepancy is 2ab. For n = 3 the discrepancy is 3a^2b + 3ab^2 — the binomial theorem catalogues all these missing terms systematically. See Why Can I Distribute Multiplication Over Addition but Not Exponents Over Addition? for the longer story.

The companion identity — difference of squares

While we are here, a second identity shows up constantly and is easy to confuse with the first.

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

Same middle term, opposite sign. The expansion is (a - b)(a - b) = a^2 - ab - ab + b^2. Two -ab pieces, not two +ab pieces.

And one more — the difference of squares (not the square of a difference):

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

This one does have only two terms, because the two middle pieces cancel: (a - b)(a + b) = a^2 + ab - ab - b^2 = a^2 - b^2. The reason students get confused is that this genuinely two-term identity looks like it gives permission to drop middle terms in general. It doesn't — the cancellation is specific to the (a-b)(a+b) pattern, and it cancels because the middle signs are opposite, not because middle terms are decorative.

The back-up rule

The title of this article is the rule. If you ever wrote (a+b)^2 = a^2 + b^2, back up and try again. Do not patch the answer, do not apologise — rewind to the step and expand (a+b)(a+b) honestly. The 2ab will appear automatically and the rest of the problem will sort itself out.

This one habit — backing up whenever a squared-sum expansion looks suspiciously short — will save you more marks over JEE-level problems than almost any other single correction. A squared sum with only two terms is, nine times out of ten, a dropped middle term. Check, back up, re-expand.

Related: Operations and Properties · Why Can I Distribute Multiplication Over Addition but Not Exponents Over Addition? · Distributive Property as an Area: a(b + c) Is Literally Two Rectangles · Read an Expression as a Tree, Not a Sentence