Formulae (a+b)², (a−b)², a²−b²

Three formulae you will meet a thousand times

(a + b)² — what is that? Watch, it is not a² + b². Try numbers: (3 + 4)² = 7² = 49, but 3² + 4² = 9 + 16 = 25. They do not match. Somewhere 24 got lost.

Where? (a + b)² means (a + b)(a + b). Expand each with each: a² + ab + ba + b² = a² + 2ab + b². The lost piece is 2ab — with our numbers 2 · 3 · 4 = 24. It fits.

Because this calculation repeats all the time, it is remembered as a formula. Today you will learn three:

(a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b).

They are not magic to cram — you can derive each in a minute by expanding. But anyone who knows them by heart calculates three times faster and, most of all: recognises them backwards too, which will help with equations.

Adam is calculating 21² in his head and it takes him a while. Tereza: “Split it. 21 is 20 + 1. Formula: (20 + 1)² = 20² + 2 · 20 · 1 + 1² = 400 + 40 + 1 = 441.” Adam tries 19² himself: “That is (20 − 1)²… 400 − 40 + 1 = 361.” Then he thinks of swapping: “And what about 21 · 19?” Tereza smiles: “(20 + 1)(20 − 1) = 20² − 1² = 399. The third formula.” Adam writes all three on a bookmark in his notebook.

Where the formulae come from and how to use them

Deriving (a + b)²: expand (a + b)(a + b) each with each: a · a + a · b + b · a + b · b. The terms ab and ba are the same, they collect: a² + 2ab + b².

Deriving (a − b)²: the same, only with a minus: a² − ab − ba + b² = a² − 2ab + b². The outer terms have plus (minus times minus), the middle one minus.

Deriving a² − b²: expand (a + b)(a − b): a² − ab + ba − b². The middle terms cancel: a² − b². It is called the difference of squares.

Using forwards — fast squaring: (x + 5)² = x² + 2 · x · 5 + 5² = x² + 10x + 25. You put a, b into the template, xong.

Using backwards — factorising: you see x² − 49 and you recognise the difference of squares: x² − 7² = (x + 7)(x − 7). This “recognising” is priceless with equations and cancelling.

First always decide what a is and what b is. With (3x + 2)², a = 3x, b = 2 — and then a² = 9x², not 3x².

Example 1: (x + 6)²

Template: a² + 2ab + b², where a = x, b = 6.

a² = x². 2ab = 2 · x · 6 = 12x. b² = 36.

Result: x² + 12x + 36.

Check by expanding: (x + 6)(x + 6) = x² + 6x + 6x + 36 = x² + 12x + 36. It fits. The formula only skipped the mid-step — the result is the same as honest “each with each”.

Example 2: (2x − 3)²

Template: a² − 2ab + b², where a = 2x, b = 3.

a² = (2x)² = 4x². Watch: the two is squared too!

2ab = 2 · 2x · 3 = 12x. With the minus sign: −12x.

b² = 9.

Result: 4x² − 12x + 9.

The most common mistake is a² = 2x². The whole a is 2x, so a² = 2x · 2x = 4x².

Example 3: 102 · 98 in your head

It looks like a job for a calculator. But: 102 = 100 + 2 and 98 = 100 − 2.

So 102 · 98 = (100 + 2)(100 − 2) = difference of squares = 100² − 2² = 10,000 − 4 = 9996.

In your head, in five seconds. This trick always works when you multiply two numbers equally far from a round one: 21 · 19, 45 · 35, 103 · 97. Formulae are not just for tests — they save work.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Practise squares up to twenty — in the formulae you need them in a flash: a², b², 2ab.

Practise the questions

On paper

Derive each formula once, then use it three times. Deriving is insurance against forgetting.

1. Derive (a + b)² by expanding (a + b)(a + b). Then from the template calculate (x + 4)² and (x + 10)².

2. Calculate (x − 5)² and (3x − 1)². For the second one first write what a is and what b is.

3. Factorise using the difference of squares: x² − 25, x² − 81, 4x² − 9.

4. Calculate in your head using the formulae: 31², 29², 31 · 29. Write next to each which formula you used.

Now you 💪

1. I can write all three formulae and derive them by expanding.
2. I do not forget the middle term 2ab.
3. I recognise the difference of squares backwards too: x² − 36 = (x + 6)(x − 6).

What to take from this lesson

• (a + b)² = a² + 2ab + b² — the middle term must not get lost.
• (a − b)² = a² − 2ab + b² — the minus is only in the middle.
• a² − b² = (a + b)(a − b) — the difference of squares.
• When a = 2x, then a² = 4x². The whole a is squared.
• You can derive each formula in a minute by expanding.

© 2026 Ing. Martin Polak / AlgoRhino · Điều khoản sử dụng nội dung