In short

The single most expensive mistake in school algebra is writing (a+b)^2 = a^2 + b^2. The correct identity is (a+b)^2 = a^2 + 2ab + b^2 — bigger by exactly 2ab. Drawn as a square of side a+b cut into four pieces, the wrong answer keeps only the two corner squares (a^2 and b^2) and throws away the two side rectangles (each of area ab). The wrong answer is not slightly off; it has a literal hole in it. Why: the squaring operator does not distribute over a sum. (a+b)^2 \neq a^2 + b^2 for the same reason \sqrt{a+b} \neq \sqrt{a} + \sqrt{b} — addition isn't squareable.

If a teacher in India has marked even one batch of class 8 algebra papers, they have seen this mistake at least fifty times: a student writes (a+b)^2 on the left, and on the right writes a^2 + b^2. It feels right. It looks tidy. The square in (a+b)^2 "distributes" onto each term — same way the minus sign in -(a+b) distributes to give -a - b. So why not?

Because squaring is not subtraction. Squaring is multiplying a thing by itself. And when you multiply a sum by itself, every part of the first sum has to meet every part of the second sum. The cross terms — the ones that get missed — are the entire reason the identity has a 2ab in it.

The fastest way to see this is to draw both sides of the wrong equation as pictures and look at what is missing.

Both pictures, side by side

Side-by-side comparison of (a + b) squared and a squared plus b squared, showing the missing 2ab areaTwo squares of the same outer size shown next to each other. The left square, labelled (a + b) squared, is divided into four pieces — a squared in the top-left, b squared in the bottom-right, and two ab rectangles in the top-right and bottom-left, each shaded a soft colour. The right square, labelled a squared plus b squared, has the same a squared and b squared corner pieces, but the two side rectangles are shaded red and labelled MISSING, showing a literal hole equal to 2ab in area. a² ab ab b² a b a b (a + b)² = a² + 2ab + b² a² MISSING ab MISSING ab b² a b a² + b² (the WRONG answer)
Same square on both sides, side $a+b$. Left: the four real pieces — $a^2$, $b^2$, and *two* $ab$ rectangles — that add to $(a+b)^2$. Right: the popular wrong answer $a^2 + b^2$ keeps only the two corner squares and leaves the two side rectangles (shaded red, marked MISSING) blank. The hole in the right picture has area exactly $2ab$ — that is the cross term the wrong answer drops.

The two pictures occupy the same outer square — both have side a+b. The left one is fully filled. The right one has a literal hole, shaped like an L wrapped around two opposite corners. That hole's area is 2ab. So:

(a+b)^2 \;-\; (a^2 + b^2) \;=\; 2ab

Not zero. Never zero, unless a=0 or b=0. Why: the squaring operator doesn't distribute over a sum. Distributivity is k(x+y) = kx + ky — a property of multiplication over addition. Raising a sum to a power is not multiplication by a fixed thing; it is multiplying the sum by itself, which forces every part to meet every part, producing cross terms.

Three worked examples

Example 1: Numbers — the gap is exactly $2ab$

Pick a = 3 and b = 4. Compute both sides.

Truth: (3 + 4)^2 = 7^2 = 49.

Wrong answer: 3^2 + 4^2 = 9 + 16 = 25.

Gap: 49 - 25 = 24.

Now compute 2ab = 2 \times 3 \times 4 = 24. The gap is exactly the missing cross term. Not approximately, not "about right" — exactly.

Numerical gap of 24 between (3+4) squared and 3 squared plus 4 squaredTwo square outlines of side seven, side by side. The left square is divided into nine, sixteen, twelve, and twelve labelled regions adding to forty-nine. The right square is divided only into nine and sixteen with the two twelve regions shaded red and labelled MISSING twelve, showing a hole of area twenty-four. 9 12 12 16 total = 49 (3 + 4)² 9 −12 −12 16 total = 25 (hole = 24) 3² + 4²
The "wrong answer" loses 24 out of 49 — almost half the square. The two missing rectangles together account for nearly as much area as the two corner squares combined.

If a and b are close in size, the missing 2ab is close to half the correct answer. The wrong version isn't a small slip; it can throw away nearly half the value.

Example 2: Why the mistake happens — FOIL it out

Squaring something means multiplying it by itself. So (a+b)^2 = (a+b)(a+b). Apply the distributive law (FOIL):

(a+b)(a+b) = \underbrace{a \cdot a}_{a^2} + \underbrace{a \cdot b}_{ab} + \underbrace{b \cdot a}_{ab} + \underbrace{b \cdot b}_{b^2}

Four products, not two. The two cross products a \cdot b and b \cdot a are equal (multiplication is commutative), so they collapse into 2ab:

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

The reason the mistake a^2 + b^2 feels natural is that students unconsciously generalise from rules like -(a+b) = -a - b or 2(a+b) = 2a + 2b, where an outside operator does distribute. But those work because multiplying by -1 or by 2 is a single multiplication. Squaring is two multiplications glued together, and the second one drags in cross terms.

Why: distributivity says k \cdot (a+b) = k \cdot a + k \cdot b for any fixed multiplier k. But in (a+b)^2 the "multiplier" is the same sum (a+b), so the sum has to multiply the sum — every term against every term — and you cannot avoid the cross products.

A useful test: any time you are tempted to "distribute" a power onto a sum, ask whether the operation outside is just one multiplication. If it is squaring, cubing, square-rooting, or any other power, the answer is no, and cross terms are coming.

Example 3: The Pythagoras trap

Some students half-recognise the equation a^2 + b^2 = c^2 from a right-angled triangle and accidentally smuggle it in here. They write (a+b)^2 = a^2 + b^2 and think they have applied Pythagoras.

They have not. Pythagoras says: if a right-angled triangle has legs of length a and b and hypotenuse c, then a^2 + b^2 = c^2. It is a statement about three lengths in a triangle, not about the square of a sum. The c is not equal to a + b. In fact, c < a + b always, by the triangle inequality.

Plug in a = 3, b = 4: Pythagoras gives c = 5, so the hypotenuse squared is 25. But (a+b)^2 = (3+4)^2 = 49, not 25. The two formulas describe completely different situations:

  • (a+b)^2 = 49 — the square built on a single segment of length a+b = 7.
  • a^2 + b^2 = 25 — the sum of squares built on the two legs 3 and 4, equal to the square on the hypotenuse 5.

The picture for Pythagoras is two squares hanging off the legs of a right triangle. The picture for (a+b)^2 is one big square cut into four pieces. Different diagrams, different theorems, no overlap.

If you ever feel the urge to write (a+b)^2 = a^2 + b^2 "because Pythagoras", stop. Pythagoras has no left-hand side that looks like (a+b)^2.

A cricket way to think about it

Imagine your team needs 7 runs from the last over. The captain is choosing a strategy.

Squaring a sum is like Plan B. The "boundaries" are a^2 and b^2 — clean, isolated efforts. But (a+b)^2 also pays for the cross-traffic between a and b — the two ab "running between the wickets" rectangles. Drop them and you've costed only the boundaries, missed all the runs scored on the move.

Numerically: imagine a and b are two batters' boundary contributions of 4 and 3. The "isolated efforts" total 4^2 + 3^2 = 25. But the full partnership — including all the strike rotation and overlapping plays between them — is (4+3)^2 = 49. The extra 24 is the two batters' interaction, not their solo scores.

Real partnerships, like real algebra, always have cross terms.

The general lesson — squaring doesn't distribute

The mistake (a+b)^2 = a^2 + b^2 is one member of a much bigger family of false distribution errors. They all look tempting and they are all wrong:

The only operations that genuinely distribute over a sum are multiplication by a fixed thing and negation. Anything else — powers, roots, reciprocals, trig functions, logs, exponentials — produces cross terms or correction terms when applied to a sum. Internalise this, and a huge slice of "silly mistakes" disappears from your algebra forever.

Once you have seen the missing red rectangles in the side-by-side picture, the urge to drop them dies down. The wrong answer literally has a hole in it. Your hand learns to fill the hole with 2ab, and the identity (a+b)^2 = a^2 + 2ab + b^2 stops being a formula to memorise — it becomes a square you complete.

References

  1. Wikipedia: Square of a sum — discussion of binomial expansion and the cross term.
  2. Wikipedia: Freshman's dream — the formal name for the mistake (a+b)^n = a^n + b^n.
  3. NCERT Class 8 Mathematics, Chapter 9: Algebraic Expressions and Identities — the standard Indian school treatment of the identity, with the same area diagram.
  4. Khan Academy: Squaring binomials of the form (x + a)² — short video on why the cross term must be there.
  5. Cut the Knot: (a+b)^2 visual proofs — a small gallery of dissection diagrams that all show the missing rectangles.