There is a mistake so common that teachers have a name for it — the freshman's dream. It goes like this. You see (a+b)^2, you notice the little 2 hanging up top, and your brain whispers "square each thing." You write:
It looks plausible. The squaring symbol sits right there, begging to be distributed. And the rule does work when you multiply: (ab)^2 = a^2 b^2. Why shouldn't it work for a sum?
Because it does not. The correct identity is (a+b)^2 = a^2 + 2ab + b^2, and the term you dropped — 2ab — is not small, not optional, not a rounding error. For a = 3 and b = 2, the true answer is 25; the wrong answer is 13; the gap is 12, which is 2 \cdot 3 \cdot 2. Almost half the answer is the missing term.
This widget makes the missing term impossible to ignore. It draws a square whose side is a + b, then slices it into the four pieces it is actually made of: one a^2 square, one b^2 square, and two ab rectangles. The two rectangles are the 2ab. They are right there in the picture. Once you see them, you cannot un-see them.
The widget
The big square has side a + b, so its area — by definition — is (a+b)^2. But look at how it is built: a horizontal cut at height a and a vertical cut at width a divide it into four pieces. The top-left blue piece is a square of side a, area a^2. The bottom-right green piece is a square of side b, area b^2. Those are the two pieces a careless student remembers. The top-right and bottom-left orange pieces are rectangles of sides a and b — each has area ab, and there are two of them, contributing 2ab. If you drop those two rectangles, you are not describing the big square any more; you are describing only about half of it.
Try these
Walk the sliders through a few configurations and watch the gap between the right answer and the wrong answer.
- a = 3, b = 2 (default). (3+2)^2 = 25; a^2 + b^2 = 9 + 4 = 13; missing 2ab = 12. The missing term is almost as big as the wrong "answer" itself. Anyone who says the two are close is fooling themselves.
- a = 4, b = 4. (4+4)^2 = 64; a^2 + b^2 = 16 + 16 = 32; missing = 32. When a = b, the two rectangles together have the same area as the two squares. The freshman's dream is off by a factor of two — you lose half the answer.
- a = 5, b = 1. (5+1)^2 = 36; a^2 + b^2 = 25 + 1 = 26; missing = 10. When one side is much smaller, the error looks small in the picture (thin orange strips) but is still a real 2ab = 10. Small b does not mean zero error.
- a = 6, b = 6. (6+6)^2 = 144; a^2 + b^2 = 72; missing = 72. Half of the true answer sits in the two rectangles you would have thrown away. It is hard to think of this as a "minor" mistake.
- a = 1, b = 1. (1+1)^2 = 4; a^2 + b^2 = 2; missing = 2. Even in the smallest case — where both sides are a single unit — the two rectangles still account for half the square. The mistake never goes away.
Why the missing term is 2ab, always
Count the orange rectangles on the canvas. There are exactly two of them — one in the top-right, one in the bottom-left. Each has one side of length a and another of length b, so each has area a \cdot b = ab. Two rectangles of area ab each add to 2ab. Not ab, not 3ab, not "it depends" — exactly 2ab.
The rectangles are not a random extra. They are what it means for a and b to combine when you square the sum. The a^2 square captures the interaction of a with itself. The b^2 square captures the interaction of b with itself. The two ab rectangles capture the cross-interactions: once for a meeting b, and once more for b meeting a. Squaring each part separately throws away these cross-terms, and the cross-terms are precisely 2ab.
The algebraic view
Every tile in the picture has a twin in the algebra. Expand (a+b)^2 by writing it as two brackets and multiplying out:
Four terms come out of FOIL, and they match the four tiles one-to-one. The first a \cdot a = a^2 is the blue square. The last b \cdot b = b^2 is the green square. The middle two, ab and ba, are the two orange rectangles — and because ab = ba, they add to 2ab. The algebra and the geometry are the same picture in two languages.
This is an identity — true for every a and every b, not just positive whole numbers. The widget only shows integer sides because you cannot easily draw a rectangle of side \tfrac{1}{2} or \pi, but the algebra carries the picture into every number you will ever plug in.
The same mistake in different clothes
(a+b)^2 = a^2 + b^2 is the most famous example of a much larger pattern. Whenever a function f gets applied to a sum, students want to distribute it. Almost always, that is wrong.
- Cubes. (a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3, not a^3 + b^3. Six extra terms hide in there (three of a^2 b and three of ab^2 collected into the middle two). The three-dimensional picture is a cube of side a+b cut into eight pieces — one a^3 cube, one b^3 cube, three a^2 b slabs, and three ab^2 slabs.
- Square roots. \sqrt{a+b} \neq \sqrt{a} + \sqrt{b}. Try a = 9, b = 16. The left side is \sqrt{25} = 5. The right side is 3 + 4 = 7. Not even close.
- Trigonometry. \sin(a+b) \neq \sin a + \sin b. The correct angle-sum formula is \sin(a+b) = \sin a \cos b + \cos a \sin b — another cross-term story, just wearing a different hat.
- Reciprocals. \dfrac{1}{a+b} \neq \dfrac{1}{a} + \dfrac{1}{b}. Test with a = b = 1: \tfrac{1}{2} versus 2.
A function f that happens to satisfy f(a+b) = f(a) + f(b) for all inputs is called a linear function, and those functions are rare enough that mathematicians gave them a special name. f(x) = 3x is linear. f(x) = x^2 is not. f(x) = \sin x is not. The rule almost always fails; linearity is the exception, not the default.
When people write (a+b)² = a² + b² mistakenly
The trap is usually the brackets. Compare these three expressions carefully:
- (a+b)^2 — the whole sum is squared. Answer: a^2 + 2ab + b^2.
- a + b^2 — only b is squared. Answer: a + b^2, already simplified.
- a^2 + b^2 — each is squared separately, then added. Answer: a^2 + b^2, also already simplified.
All three are valid expressions. Only the first one has a rewrite, and that rewrite includes 2ab. The mistake is treating the first expression as if it were the third. The fix is to read the brackets before you do anything. If the ^2 sits outside a bracket, the whole bracket is being multiplied by itself — you must FOIL, and the cross-term 2ab will show up every time.
The identity you should memorise
Two identities, together, cover most of the squaring you will ever do in Class 9, Class 10, and all the way into JEE problems.
The sign on the cross-term flips when the bracket has a minus, but the 2ab is still there — you do not get to drop it. A third, related identity is (a+b)(a-b) = a^2 - b^2, the difference of squares, which has no cross-term because the plus-and-minus versions cancel out. The widget's picture is also the quickest way to see why (a-b)^2 = a^2 - 2ab + b^2: start with the big a^2 square, remove two ab rectangles (that is -2ab), and notice that a b^2 square got removed twice, so you have to add one back — leaving a^2 - 2ab + b^2.
Once the tile picture is in your head, the identities stop being strings of symbols to memorise. They are shapes. Two squares and two rectangles for (a+b)^2. Two rectangles you cannot wish away. The next time your pen starts to write (a+b)^2 = a^2 + b^2, you should feel the orange tiles tugging at your sleeve, reminding you that you are about to lose 2ab of area. Look at the picture, count the tiles, and write the full identity. The tiles make the missing term visible; once you have seen the two rectangles, you cannot un-see them.