Difference of Squares and the Perfect Square Formula — Explained 7+

Difference of squares, seen as tiles

Take a square with side a and cut a smaller square of side b out of one corner. What is left is an L-shaped piece with area a² − b². Now make one straight cut across the L and slide the piece you cut off around to the other end: the parts fit together into a rectangle with sides (a − b) and (a + b). Nothing was added or thrown away, so the areas must be equal:

a² − b² = (a − b)(a + b)

Factoring: x² − 9 = (x − 3)(x + 3) · x² − 25 = (x − 5)(x + 5) · 4x² − 49 = (2x − 7)(2x + 7).

Mental arithmetic: 43 · 37 = (40 + 3)(40 − 3) = 1600 − 9 = 1591, and 101 · 99 = 10,000 − 1 = 9999.

Solving: x² − 16 = 0 becomes (x − 4)(x + 4) = 0, so x = 4 or x = −4.

Two warnings. First, a sum of squares, a² + b², does not factor over the real numbers — that is a property of the expression, not a gap in your memory. Second, "difference of squares" (a² − b²) and "square of a difference" ((a − b)²) are different animals; the names are nearly identical and the expressions are not.

The perfect square formula: why the middle term is 2ab

Draw a square with side a + b and split each side into a piece of length a and a piece of length b. The square falls apart into four tiles:

• the big square a × a, which is a²;
• the small square b × b, which is b²;
• and two identical rectangles a × b, one along the top and one along the side. Those give 2ab.

Check it with numbers: side 3 + 2 = 5, total area 25, and the tiles give 9 + 6 + 6 + 4 = 25. This is also why the classic error (a + b)² = a² + b² is so easy to catch: 9 + 4 = 13, not 25. Two rectangles are missing.

Binomial square examples: (x + 4)² = x² + 8x + 16 · (2x + 3)² = 4x² + 12x + 9. Mental arithmetic: 51² = (50 + 1)² = 2500 + 100 + 1 = 2601.

Square of a difference: where the +b² comes from

The formula (a − b)² = a² − 2ab + b² looks odd — you subtract twice and then suddenly add. The picture explains it. Start from a square of side a. To shrink it to side a − b you shave a strip of width b off the top and another off the side: that is the −2ab. But the little b × b corner lies where the two strips overlap, so it has been removed twice. Putting it back once is the +b².

Examples: (x − 5)² = x² − 10x + 25 · 99² = (100 − 1)² = 10,000 − 200 + 1 = 9801.

Note the sign pattern: only the middle term flips sign with the bracket. The outer terms stay positive, because a square is never negative.

Cube of a binomial

(a + b)³ = a³ + 3a²b + 3ab² + b³, and it has a picture too, just a solid one: a cube of side a + b splits into eight blocks — one cube a³, three flat slabs a²b, three bars ab², and one small cube b³. The 3s in the formula are simply "three of that block."

The cube of a difference alternates signs: (a − b)³ = a³ − 3a²b + 3ab² − b³.

Example: (x + 2)³ = x³ + 6x² + 12x + 8.

Sum and difference of cubes

These two factor where you might not expect it:

• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)

The first bracket carries the same sign as the original; the middle term of the second bracket carries the opposite one. The second factor looks like a perfect square but has a single ab rather than 2ab, which is why it is called an incomplete square. Example: x³ − 8 = (x − 2)(x² + 2x + 4).

The algebraic identities worth knowing

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• a² − b² = (a − b)(a + b)
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a − b)³ = a³ − 3a²b + 3ab² − b³
• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)

They earn their keep in three places: expanding brackets in one step, factoring so that fractions cancel — (x² − 4) / (x + 2) = (x − 2)(x + 2) / (x + 2) = x − 2 — and squaring awkward numbers in your head. In the US curriculum they turn up in Algebra 1 and stay useful all the way through calculus.

How the game teaches them

In the game above the child is the master mason of a fortress. Every section runs in three phases. Order: pick slabs at the depot — that is expanding the brackets, written as an invoice. Laying: place the slabs with no overlaps and no gaps, checked by a real tiling algorithm rather than by eye. Assault: solid masonry holds; any open cell becomes the breach, and the debrief names the missing piece and its exact size.

The trap that makes the lesson stick is the depot's "Beginner's kit": just two squares, the big one and the small one. Order it and the wall keeps two rectangular holes — and the raiders climb through those holes and nowhere else. Those holes are 2ab. The child does not get a red cross; the child watches the consequence.

A hundred sections across ten regions follow the same order as this page:

• 1–15 — square of a sum with concrete numbers, no letters at all;
• 16–30 — bigger numbers, the guide fades out, the Beginner's kit trap appears;
• 31–45 — the same walls, now labelled with a and b;
• 46–60 — difference of squares: the forge cuts a damaged slab and folds it into a clean rectangle;
• 61–75 — square of a difference: the wall shrinks and the corner square has to be put back;
• 76–88 — mixed waves, and from section 79 the order is placed blind, computed in the head;
• 89–100 — cube of a sum: a donjon assembled from eight blocks.

Time is paused while the child thinks. There is no way to lose — only to rebuild the wall.