In short

The identity (a - b)^2 = a^2 - 2ab + b^2 is the cut out corners then add back overlap pattern. Start with a square of side a. Slice off a strip of width b from the right and another from the bottom — that subtracts 2ab. But the small b \times b corner where the two strips overlap got subtracted twice, so add one b^2 back. What remains is a square of side a - b, with area a^2 - 2ab + b^2.

You already know the picture for (a + b)^2 — a square of side a + b chopped into four neat tiles. (a - b)^2 has its own picture, and once you see it, the minus sign in the middle stops being a thing you have to remember. It just shows up because you are subtracting strips rather than adding them. And the +b^2 at the end? That is the geometric "oops, I removed the corner twice" correction.

The construction

Start with a big square of side a. Its area is a^2. You want a square of side a - b sitting inside it. So you need to remove the L-shaped border that wraps around the bottom and right.

Here is the natural way to remove that L: slice off a vertical strip of width b from the right side, then slice off a horizontal strip of width b from the bottom. Each strip has area a \times b = ab. Total removed so far: 2ab.

But look at the bottom-right corner. The vertical strip and the horizontal strip overlap there in a small b \times b square. You removed that little corner once when you took out the vertical strip, and again when you took out the horizontal strip. Why: the same square cannot be subtracted twice and still vanish — once it is gone, it is gone. The second subtraction removed area that was no longer there, so you owe that b^2 back to the picture. Add b^2 once to undo the double-counting.

What is left? Exactly the top-left square of side a - b. Its area:

\text{area} = a^2 - \underbrace{ab}_{\text{right strip}} - \underbrace{ab}_{\text{bottom strip}} + \underbrace{b^2}_{\text{corner added back}} = a^2 - 2ab + b^2

And since the leftover region is the (a - b) \times (a - b) square, that area also equals (a - b)^2. Two ways of measuring the same shape:

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

The picture is the proof.

Geometric proof of (a minus b) squaredA square of side a. The right vertical strip of width b is shaded as the first subtraction. The bottom horizontal strip of width b is shaded as the second subtraction. The bottom-right corner of side b is shaded more deeply because it has been subtracted twice and must be added back once. The remaining top-left region is the square of side a minus b with area a squared minus 2ab plus b squared. (a−b)² −ab −ab +b² a − b b a − b b Start: a² (whole square) Subtract right strip: −ab Subtract bottom strip: −ab Add back overlap: +b² a² − 2ab + b² = (a−b)²
Start with the $a \times a$ square. Remove the right strip (area $ab$). Remove the bottom strip (area $ab$). The dark $b \times b$ corner was subtracted by *both* strips, so add it back once. What remains is the $(a - b) \times (a - b)$ square.

The light-shaded strips show what you removed. The deeper-shaded b^2 square shows the over-subtracted corner that has to be paid back. Why two strips and one corner-back, not just an L? You could compute the L-shaped border's area directly, but it is awkward — an L is not a rectangle. Splitting into two clean rectangles is easier; the price is one over-counted corner, which is trivial to fix.

Worked examples

Example 1 — the picture with real numbers

Take a = 10 and b = 3. The big square has area 100. The right strip is 10 \times 3 = 30. The bottom strip is also 30. The corner is 3 \times 3 = 9.

10^2 - 2(10)(3) + 3^2 = 100 - 30 - 30 + 9 = 49

And the leftover square has side 10 - 3 = 7, with area 7^2 = 49. ✓ The two count match.

a equals 10 and b equals 3 worked exampleA 10 by 10 square divided into a 7 by 7 inner square, two 7 by 3 strips, and a 3 by 3 corner. The strips are lightly shaded as subtractions and the corner is darker as the overlap added back. 7² = 49 −30 −30 +9 7 3
$100 - 30 - 30 + 9 = 49 = 7^2$. The arithmetic and the geometry agree.

Example 2 — mental maths for $97^2$

You want 97^2 in your head. Write 97 = 100 - 3 and use (a - b)^2 = a^2 - 2ab + b^2 with a = 100, b = 3:

97^2 = (100 - 3)^2 = 100^2 - 2(100)(3) + 3^2 = 10000 - 600 + 9 = 9409

Three steps, all easy: ten thousand, subtract six hundred, add nine. No long multiplication. The same trick crushes 98^2 = 10000 - 400 + 4 = 9604, 999^2 = 1{,}000{,}000 - 2000 + 1 = 998001, and 48^2 = 2500 - 200 + 4 = 2304 (using a = 50, b = 2).

A cricket scoring example: the team needs 97 runs to win and you want to know what 97 \times 97 is for some imaginary stat. Same trick. 9409. You did not reach for a calculator.

Example 3 — the sign of the middle term, side by side

Lay the two square identities next to each other:

Identity Geometric move Middle term
(a + b)^2 = a^2 + 2ab + b^2 Add two ab strips around the a^2 square +2ab
(a - b)^2 = a^2 - 2ab + b^2 Remove two ab strips from the a^2 square -2ab

Same two strips. Different direction. Why is the corner +b^2 in both identities? In (a+b)^2 the b^2 tile fills the empty corner you created by adding strips. In (a-b)^2 the b^2 pays back the corner you removed twice. Different reason, same sign — the b^2 is always added. That is why the last term never carries a minus.

So when a student writes (a - b)^2 = a^2 - 2ab - b^2 on a test, the picture says no: the corner is added back, not removed again.

Why this matters

Once the picture is in your head, you stop confusing the two identities and you stop forgetting which sign goes where. You also pick up a free mental-arithmetic skill: any number of the form (round number minus a small bit) squares quickly. 19^2? 400 - 40 + 1 = 361. 29^2? 900 - 60 + 1 = 841. 199^2? 40000 - 400 + 1 = 39601.

And the idea — "subtract too much, then add back the overlap" — is the seed of inclusion–exclusion, one of the most useful counting principles in mathematics. The same logic that fixes the double-counted corner here fixes double-counted students in a Venn diagram, double-counted multiples in number theory, and double-counted intersections in probability. You already know inclusion–exclusion. You learnt it by squaring 97.

References

  1. Algebraic identities — the parent article.
  2. Wikipedia: Square of a binomial
  3. NCERT Class 8 Mathematics, Chapter 9: Algebraic Expressions and Identities — official syllabus treatment.
  4. Cut the Knot: Proofs without words — collection of geometric proofs.
  5. Wikipedia: Inclusion–exclusion principle — the same "subtract twice, add back once" idea, generalised.