In short

This is the sister error to (a+b)^2 = a^2 + b^2. The wrong instinct is the same: see the little 2 outside the bracket, drop it onto each term, smile, move on. So (a-b)^2 becomes a^2 - b^2 and you lose two marks. Worse, this version has a second layer of bugs hidden inside it. Some students do remember the cross term but write a^2 + 2ab + b^2 — they have remembered the 2ab and forgotten the minus sign in front of it. Three corrections kill both versions: (1) the middle term is NEGATIVE because we are subtracting, (2) test with numbers — a=5, b=3 is enough, and (3) FOIL out loud, paying full attention to every minus sign. In CBSE class 9 algebra, this is one of the top mistakes on the unit-test. Why: the sign isn't a cosmetic detail — it comes from a real -ab produced by multiplying a by -b inside the bracket. Track the sign at the source and the bug never enters your work.

You probably know the correct identity by heart: (a-b)^2 = a^2 - 2ab + b^2. You have written it in the chapter notes. Your maths teacher has chanted it. NCERT has printed it. And yet, in the middle of a CBSE Class 9 unit-test on Algebraic Identities, your hand will quietly write (x-3)^2 = x^2 - 9 and your evaluator will quietly cut two marks. Again.

This article is the subtraction-side companion to (a+b)² = a² + b². The cognitive disease is identical — your pattern-matcher confuses an exponent with a multiplier — but the cure has an extra ingredient. The minus sign. Most students who fix the "drop-the-cross-term" bug then run straight into a second one: writing +2ab when the correct middle term is -2ab. Both bugs are killable. Both kills require a habit, not just a fact.

The two bugs, stacked

The wrong identity comes in three flavours, and you should be able to recognise yourself in at least one.

Three traps in the (a-b) squared identity, stackedA vertical stack of three boxes. The top box, marked with a red cross, shows (a-b) squared equals a squared minus b squared, labelled "Trap 1: drop the cross term". The middle box, also red cross, shows (a-b) squared equals a squared plus 2ab plus b squared, labelled "Trap 2: keep the cross term but get the sign wrong". The bottom box, marked with a green tick, shows (a-b) squared equals a squared minus 2ab plus b squared, labelled "Correct". TRAP 1 — drop the cross term entirely (a − b)² = a² − b² ✗ TRAP 2 — remember the cross term, get the sign wrong (a − b)² = a² + 2ab + b² ✗ CORRECT — minus sign on the middle term (a − b)² = a² − 2ab + b² ✓ the −2ab is what makes it the *square of a difference*, not a sum
Three versions of the same identity. Two are wrong, one is right. Trap 1 is the same pattern-match disease as the $(a+b)^2$ mistake — your hand drops the cross term. Trap 2 is sneakier: the cross term is back, but its sign has flipped from $-$ to $+$, leaving you with the expansion of $(a+b)^2$ instead of $(a-b)^2$. Both wrong answers are wrong by exactly $4ab$ and $2ab$ respectively — large enough to wreck any further calculation.

Trap 2 is the part that does not appear in the addition-side article, and it is the one that catches the better students — the ones who memorised the formula but did not internalise where the minus sign comes from. The fix is the same in both cases: stop reciting the formula, and start deriving the middle term every time, paying attention to where each sign is born.

Why the bugs survive

Bug 1 is exactly the addition-side mistake: your brain treats the little 2 as a multiplier and pushes it onto each term. The cure is the rectangle picture and the FOIL discipline.

Bug 2 has its own cause. When you memorise (a-b)^2 = a^2 - 2ab + b^2 as a string of symbols, you remember the digits — a^2, 2ab, b^2 — and lose the signs. Your handwriting fills in plus signs by default because plus signs are the most frequent connector in algebra. The minus sign on 2ab has to be earned by the derivation, and if you skip the derivation you skip the earning.

In CBSE Class 9 Algebra, examiners report this bug constantly. It costs marks not just on the identity question itself but on every downstream step — completing the square, factorising perfect-square trinomials, simplifying expressions involving (a-b)^2 inside larger problems. One sign error in line 2 corrupts every line below it.

Correction 1: The middle term is NEGATIVE because we are subtracting

The minus sign on 2ab is not a memorisation challenge. It is a consequence. Here is where it is born.

(a-b)^2 means (a-b)(a-b). Multiply every term in the first bracket by every term in the second:

(a-b)(a-b) = a \cdot a + a \cdot (-b) + (-b) \cdot a + (-b) \cdot (-b)

Look at each product carefully:

Add them up:

a^2 - ab - ab + b^2 = a^2 - 2ab + b^2

Why: the two middle products are both negative because each one is a positive number times a negative number. The last term (-b)(-b) is positive because two negatives cancel. There is no way to do this multiplication slowly and end up with +2ab — the minus sign is forced by the rules of signed multiplication.

The minus on 2ab is not a stylistic choice. It is what happens when -b meets a twice. Once you have seen this happen on the page, the wrong sign feels mechanically impossible.

Correction 2: Test with numbers

Numbers cannot lie. Pick a small a and b and check.

The number test that kills both bugs at once

Suppose, on a CBSE Class 9 unit-test, you have written (a-b)^2 = a^2 - b^2. Stop and test with a = 5, b = 3.

Left side: (5 - 3)^2 = 2^2 = 4.

Right side: 5^2 - 3^2 = 25 - 9 = 16.

4 \neq 16. The two sides disagree by 12 — a huge gap. Whatever you wrote down is wrong.

Number test of the wrong identity (a-b) squared equals a squared minus b squaredTwo boxed expressions side by side. The left box shows (5-3) squared equals 4. The right box shows 5 squared minus 3 squared equals 16. A red cross sits between them indicating they disagree. left side (5 − 3)² = 4 claimed right side 5² − 3² = 16 ✗ 4 ≠ 16 the identity is wrong by 12
One counter-example is enough to kill an identity. $(5-3)^2$ really is $4$, and $5^2 - 3^2$ really is $16$. They are not equal, so the proposed identity $(a-b)^2 = a^2 - b^2$ cannot hold for all $a$ and $b$. Three seconds of arithmetic, and you know to throw the line away and try again.

Why: an identity is a claim about every pair of values. A single mismatched pair is enough to disprove it. You don't need the cleverest numbers — a=5, b=3 exposes nearly every false identity in this chapter.

Now run the same test on the correct identity to feel the contrast.

The number test that confirms the right answer

Same a = 5, b = 3. This time check (a-b)^2 = a^2 - 2ab + b^2.

Left side: (5 - 3)^2 = 2^2 = 4.

Right side: 5^2 - 2(5)(3) + 3^2 = 25 - 30 + 9 = 4.

Both sides equal 4. The identity holds.

Notice what the -30 is doing. It is the -2ab = -2 \times 5 \times 3 = -30 term. Without that minus sign, the right side would be 25 + 30 + 9 = 64, which is 8^2 = (5+3)^2 — the answer to the wrong question. The minus sign is what distinguishes "square of a-b" from "square of a+b".

Why: the -2ab term is exactly what pulls a^2 + b^2 down to the correct value. With a=5, b=3: a^2 + b^2 = 34, and you must subtract 2ab = 30 to land at 4. The size of the correction is enormous — 30 out of 34. Forgetting the sign is not a small slip; it changes the answer by 60.

The number test takes three seconds and answers two questions at once: is the identity right? and if not, by how much? Run it whenever you write down a squared bracket you are unsure about.

Correction 3: FOIL out loud — every sign tracked

The third habit is to enumerate the four products with their signs, out loud or in a careful whisper. This is the same FOIL discipline as the addition case, but with extra attention to signs.

FOIL the sign trap

Take the case some students stumble on: they remember the 2ab but lose the minus. Run FOIL slowly to see exactly where the minus is born.

Whisper:

"a times a. a times minus b. Minus b times a. Minus b times minus b."

Write each one with its sign:

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

Four products. The middle two are both negative — that is where the minus sign on the cross term comes from. They add to -2ab. The last product is positive — negative times negative is positive — so the b^2 at the end is +b^2, not -b^2.

Combine:

= a^2 - 2ab + b^2

Why: every sign in the final answer is forced by the signs of the inputs. You did not choose -2ab; the multiplication chose it for you. The trick to never writing +2ab again is to say each sign aloud while you write it. Pattern-matching skips signs; enumeration cannot.

Practise on (2x - 5)^2. Whisper: "2x times 2x, 2x times minus 5, minus 5 times 2x, minus 5 times minus 5." Write: 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25. The minus sign on 20x wrote itself.

After a week of this, your hand stops needing the whisper — the sign tracking happens automatically. The wrong shortcut a^2 + 2ab + b^2 for a minus bracket starts to feel as wrong as writing (-3) \times (-3) = -9.

A geometric backup: subtract two strips, add back the corner

When the algebra still feels slippery, the picture is solid. (a-b)^2 is the area of a square of side a-b. You can build that square out of a bigger square of side a by removing two strips and adding back the corner you removed twice.

Start with a square of side a (area a^2). Cut a strip of width b from the right edge (area ab, removed). Cut a strip of width b from the bottom edge (area ab, also removed). The little corner of side b at the bottom-right has now been removed twice — once by each strip — so add it back once (area b^2, returned).

Total: a^2 - ab - ab + b^2 = a^2 - 2ab + b^2. The remaining region is exactly the square of side a-b.

Why: each strip you removed had area ab, contributing the two negative -ab pieces that combine into -2ab. The corner was double-counted in your removal, so you put it back once with a +b^2. Every sign on the right-hand side has a geometric job: minus signs are removals, plus signs are restorations. The full geometric proof lives in Geometric Proof of (a−b)²: Square Minus Corner.

This picture is the visual reflex you want for (a-b)^2. The moment your eye sees the bracket, your brain should picture the big square with two strips removed and the small corner restored. The wrong sign cannot survive that picture.

Putting them together

Three habits, three different cognitive jobs:

Use whichever is easiest in the moment. In an exam, the number test is silent and quick; FOIL keeps you honest in a long derivation; and the strip-and-corner picture is the visual gut-check that no sign can hide from. Run at least one on every (a-b)^2 you meet for two weeks. After that, the right answer becomes automatic — and writing (a-b)^2 = a^2 - b^2 or (a-b)^2 = a^2 + 2ab + b^2 will feel as physically wrong as forgetting to put the run-rate denominator under a cricket strike-rate.

References

  1. Wikipedia: Square (algebra) — algebraic background on squaring binomials and why cross terms appear.
  2. NCERT Class 8 Mathematics, Chapter 9: Algebraic Expressions and Identities — first formal Indian school treatment of (a-b)^2.
  3. NCERT Class 9 Mathematics, Chapter 2: Polynomials — the identity restated and applied in factorisation; this is where unit-test sign errors are most costly.
  4. Khan Academy: Multiplying binomials — short videos on FOIL with negative terms.
  5. Cut the Knot: Algebraic Identities — dissection proofs including the strip-and-corner picture for (a-b)^2.