The square of a binomial, cut into four.
“(a+b)² = a² + b²” is the most common mistake in all the algebra of the first two years of high school — and you can see exactly why it is wrong as soon as you draw it: a piece is missing, in fact two, and both have the same name. Here you can change a and b with two sliders and watch the four areas recalculate live.
(a + b)² = a² + 2ab + b²
The identity
The interactive figure
Use the arrows on the figure (or swipe) to build the proof one piece at a time, then drag the a and b sliders — or pick a ready-made shape — and watch the four areas and their sum recalculate. On a keyboard, ← and → change step.
Hover over (or tap) a part of the drawing — the square, a region, the sum — to find out what it is.
The mistake that (almost) everyone makes once
Many people write (a + b)² = a² + b², as if the square “distributed” over the sum. It does not work: try a = 2 and b = 3. (2+3)² is 25, but 2² + 3² is only 13. 12 are missing — and those 12 are exactly 2 · 2 · 3, the piece this figure shows.
a² = 252ab = 30b² = 9(a + b)² = 64
With a = 5 and b = 3: 25 + 15 + 15 + 9 = 64, exactly (a + b)² = 64.
The proof, step by step
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Step 1 · The mistake
The mistake that (almost) everyone makes once
Many people write (a + b)² = a² + b², as if the square “distributed” over the sum. It does not work: try a = 2 and b = 3. (2+3)² is 25, but 2² + 3² is only 13. 12 are missing — and those 12 are exactly 2 · 2 · 3, the piece this figure shows.
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Step 2 · The square
We start from a square of side a + b
a and b are two segments placed end to end: together they make the side of the big square, and the area of that square is by definition (a + b) squared — it is the same thing, drawn instead of written.
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Step 3 · The cut
One horizontal cut and one vertical cut, at the same distance
The vertical cut is a from the left side, the horizontal one a from the top side — the same a in both cases. The square is divided into four rectangles, and not by chance: they are the four possible combinations of “the a part” and “the b part”.
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Step 4 · The regions
Four regions: a², ab, ab, b²
Top left, a square of side a, area a². Bottom right, a square of side b, area b². The other two are a × b rectangles, area ab each — and there are two of them, not one: that is where the term the initial mistake was losing hides.
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Step 5 · The sum
Add them all up: a² + 2ab + b²
The two ab areas are equal, so together they make 2ab. The area of the big square — which we already knew to be (a + b)² — is the sum of the four regions: a² + 2ab + b². It is not a rule to learn by heart, it is an area calculation you can redraw every time.
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Step 6 · Try it
Change a and b and watch the identity stay true
Drag the sliders: the square is redrawn, the numbers below change, but the sum of the four areas always comes back equal to (a + b)². Try making a and b very different, or equal: the identity has no exceptions.
What each region of the drawing is
In the figure these explanations appear when you hover over the element. Here they are all written out.
- The big square
- Side a + b, area (a + b)². It is the starting point: everything else is just a different way of counting the same area.
- The square a²
- Side a, the top-left corner. Its area is a times a, that is a².
- The two ab rectangles
- Side a by side b, one above and one below the cut. They are two identical rectangles: together they are worth 2ab, the “middle” term of the identity.
- The square b²
- Side b, the bottom-right corner. Its area is b times b, that is b².
- The sum of the four areas
- a² + ab + ab + b², that is a² + 2ab + b². Always equal to the area of the big square (a + b)² — because it is the same area, just divided into four pieces.
The questions we always get.
Why is (a + b)² not a² + b²?
Because (a + b)² means (a + b) multiplied by (a + b), and that product has four terms, not two: a·a, a·b, b·a and b·b. The two mixed terms a·b and b·a are equal and add up to 2ab, which the common mistake loses completely.
Does it work with negative numbers too?
Yes: (a − b)² = a² − 2ab + b², the same identity with b replaced by −b. The sign of 2ab changes because a·(−b) and (−b)·a are negative, but the structure — a square, two equal rectangles, a square — stays the same.
What is it used for in practice, besides simplifying calculations?
It is the first example of a “special product”: a calculation shortcut you will use for years, from algebraic factorising to completing the square in quadratic equations. Anyone who learns it as a drawing, and not as a formula to remember, applies it correctly for much longer.
Keep going from here
Special products never stick?
The square of a binomial is the first of a family you will use for years — factorising, quadratic equations, radicals. Anyone who learns it as a drawing, not as a formula, never gets it wrong again.