Formulae (a+b)², (a−b)², a²−b²

Three formulae used all the time in algebra. Where they come from and how to calculate faster with them.

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.

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The middle term 2ab is “twice both together”. When you are not sure of the formula, expand (a + b)(a + b) by hand — the formula will derive itself in thirty seconds.

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, 完成.

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.

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(a + b)² is NOT a² + b². The middle term 2ab must not get lost. Check it any time with numbers: (3 + 4)² = 49, but 9 + 16 = 25. Anyone who forgets 2ab loses marks in every other year-8 question.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 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.

Derive (a + b)² by expanding (a + b)(a + b). Then from the template calculate (x + 4)² and (x + 10)².
Calculate (x − 5)² and (3x − 1)². For the second one first write what a is and what b is.
Factorise using the difference of squares: x² − 25, x² − 81, 4x² − 9.
Calculate in your head using the formulae: 31², 29², 31 · 29. Write next to each which formula you used.

Now you 💪

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

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Done when: You have a derived formula, eight uses on paper including calculating in your head, and a run of five correct in practice.

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.