Algebraic Identities – A Reference for Common Patterns

Each identity above has a matching picture. Seeing them as areas of squares and rectangles explains why they're true, not just that they're true.

(a + b)² = a² + 2ab + b²

Let's build a picture of (a + b)² so you can literally see why the formula is true. Draw one big square and make every side exactly (a + b) long.

Steps:

1. Split each side into two parts. On the top side, mark off a length a, then a length b right after it — the curly bracket over the top confirms the two parts together make (a + b). Do exactly the same down the left side.
2. Draw lines across from the split points. One line straight down from the a/b split on top, and one line straight across from the a/b split on the left. This carves the big square into four smaller pieces.
3. Work out each piece's area. The top-left piece is a × a — a square with area a². The bottom-right piece is b × b — a square with area b². The top-right and bottom-left pieces are both a × b — two matching rectangles, each with area ab.
4. Add up all four pieces. a² + ab + ab + b² = a² + 2ab + b², since the two ab rectangles combine to give 2ab.

Press each step below to build the square up piece by piece, watching the formula grow to match.

(a − b)² = a² − 2ab + b²

This time, start with a big square of side a — the curly brackets confirm the whole top and the whole left side are each a. Split the top and left edges into an (a − b) part and a b part, dividing the square into four regions: the (a − b)² piece we actually want, two b-wide/tall rectangles, and a b×b corner. Press each step below in order (or use Auto Play) to watch a² turn into a² − 2ab + b² piece by piece.

a² − b² = (a + b)(a − b)

Start with a square of side a, split into a left column (width a − b, full height a), a strip sitting above the bottom-right corner (width b, height a − b), and the corner itself, a b×b square. Press each step below to see it turn into a rectangle.

Steps:

1. Set up the pieces. The square splits into three parts: the tall left column (a − b wide, a tall), the strip above the corner (b wide, a − b tall), and the b² corner.
2. Slide the strip down. The strip turns a quarter turn and slides down until it sits directly under the left column. After turning, its width is exactly (a − b) — the same as the column — so the two fit together perfectly with no gaps.
3. See the new rectangle. The column and the turned strip now form one plain rectangle: width (a − b), height a + b (the column's a plus the strip's b).
4. Compare the areas. Nothing was added or taken away except the excluded b² corner, so the rectangle's area, (a + b)(a − b), is exactly a² − b².

Going Further: (a + b + c)²

The same idea extends to three terms. A square with side (a + b + c) splits into a 3-by-3 grid of nine regions instead of four.

Adding up all nine regions: the three "diagonal" squares a², b² and c² each appear once, while ab, ac and bc each appear twice (once above and once below the diagonal). That gives:

Going Further Still: (a + b + c + d)²

The same idea keeps extending. A square with side (a + b + c + d) splits into a 4-by-4 grid of sixteen regions instead of nine.

Adding up all sixteen regions: the four "diagonal" squares a², b², c² and d² each appear once, while every pair — ab, ac, ad, bc, bd and cd — appears twice (once above and once below the diagonal). That gives:

Your Turn: Can You Do (a + b + c + d + e)² Yourself?

You've now seen the pattern grow from four regions, to nine, to sixteen. Can you predict what happens with five terms? A square of side (a + b + c + d + e) would split into a 5-by-5 grid of 25 regions. See if you can work out: how many diagonal squares will there be? How many different cross-term pairs, and how many times does each one appear? And putting it all together — what does the full expansion of (a + b + c + d + e)² look like?

Going Further: (a + b)³

The same idea works in 3D. A cube of side (a + b) splits into eight smaller boxes instead of four flat pieces — giving (a + b)³ = a³ + 3a²b + 3ab² + b³. This one is much easier to see as an actual rotating 3D model than as a flat picture.

Explore the Cube of a Binomial →

• Mental arithmetic tricks: squaring numbers near a round number quickly, using the identities.
• Engineering formulas: many formulas are pre-simplified using these identities.
• Computer science: identities help simplify expressions before writing efficient code.