Free maths tool · Level: Ages 13 to 15
Binomial squares
The classic slip is (a+b)² = a² + b². Draw the square and what is missing becomes obvious.
For (a+b)², draw a square of side a+b. It breaks into four pieces: a², two rectangles of a by b, and b². Together that is a² + 2ab + b². Those two rectangles are exactly what gets dropped when someone writes a² + b².
Why that 2ab is really there
Set a to 5 and b to 3. The big square has side 8, so area 64. But a² + b² gives 25 + 9 = 34. Thirty is missing, which is exactly 2 lots of 15: the two amber rectangles.
So you can always catch this slip with numbers. Pick two values, work out both sides, and see whether they match.
Why (a-b)² feels harder
With the minus you start from the big a² square and take away two strips of a by b. But in the corner where those strips overlap, you have removed the same piece twice.
That corner is b². You have to add it back once, which is why the formula ends in plus b² and not minus b². Switch the tool to (a-b)² and the red corner is precisely that piece.
The difference of two squares
For (a+b)(a-b) you cut a small b² square out of the corner of the big a² square. What is left has area a² - b², and it rearranges into one rectangle of (a+b) by (a-b).
This is the most useful of the three to recognise when factorising. Given x² - 9 you can write (x+3)(x-3) without calculating anything.
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Frequently asked questions
- Why is (a+b)² not a² + b²?
- Because there are two rectangles of a by b in the middle. The square of side a+b has four pieces, not two. Together they give a² + 2ab + b².
- What are the three binomial identities?
- (a+b)² = a² + 2ab + b², (a-b)² = a² - 2ab + b², and (a+b)(a-b) = a² - b². The first two differ only in the sign of the middle term.
- Why is there a plus b² in (a-b)²?
- Because taking away both strips removes the corner piece twice. Adding it back once fixes the count, and that add-back is the plus b².
- How do I spot them when factorising?
- Look for two squares. With a minus between them it is a difference of squares. With three terms where the middle one is twice the product of the roots of the outer two, it is a perfect square.