What it does
An identity such as (a + b)² = a² + 2ab + b² is true for every a and b. The easiest way to see why is to draw it: a square of side a + b cuts into a square a², a square b² and two rectangles a × b. This page draws six identities that way, with your own numbers:
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²: two strips are cut off a², and the corner they share comes back once.
- (a + b)(a − b) = a² − b²: cut b² out of a², slide the leftover strip across, and the L shape becomes a rectangle.
- (x + a)(x + b) = x² + (a + b)x + ab
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
- (a + b)³ = a³ + 3a²b + 3ab² + b³: a cube made of eight blocks that you can pull apart.
It fits the Class 8 topic algebraic expressions and identities and the identities in the Class 9 polynomials chapter (including the three-term square and the cube).
Use these identities to factor quadratics in the quadratic explorer and to read sums and products of zeros in polynomial zeros. For bigger expressions, let symbolic algebra expand or factor them.
How to use it
- Pick an identity at the top.
- Move the sliders (or type values) for a, b, and where needed x or c. The picture redraws and every piece shows its name.
- The line under the picture puts your numbers into the identity and adds the pieces up.
- For (a + b)(a − b), use the ↔ slider to move the strip. For (a + b)³, use the ↕ slider to pull the eight blocks apart; labels appear when they are apart.
A worked example
The page starts with (a + b)² for a = 5 and b = 3.
- The big square has side 8, so its area is 64.
- The pieces: a² = 25, two strips of ab = 15 each, and b² = 9.
- 25 + 2 × 15 + 9 = 64. ✓
Switch to (a + b)(a − b) with the same numbers: 8 × 2 = 16 and 25 − 9 = 16. The cut-out corner is b² = 9, and the leftover strip 2 × 3 moves to make the rectangle 8 × 2.
Switch to (a + b)³: 512 = 125 + 3 × 75 + 3 × 45 + 27.
Tips for mental maths
- 103² = (100 + 3)² = 10 000 + 600 + 9 = 10 609.
- 98² = (100 − 2)² = 10 000 − 400 + 4 = 9 604.
- 105 × 95 = 100² − 5² = 9 975.
The Check yourself questions practise exactly these tricks, plus factoring x² + px + q and finding a² + b² from a + b and ab. Nothing you type is saved.
FAQ
Why must b be smaller than a in two of the pictures? The identities hold for any numbers, but a picture of a − b only makes sense when a − b is a length. The page tells you if b is bigger.
Is this from the NCERT book? It covers the same identities with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- The standard algebraic identities and their area and volume models are classical mathematics. Pictures and wording are fibub's own.
- Every identity checked with sympy.expand; the tiles are checked to fill each shape exactly for random sizes, and the cube's blocks to fill (a + b)³ with counts 1, 3, 3, 1.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.