A bracket, taken twice

Liri found an old star-map sketch: a square search area whose side was made of two parts – (x + 3)². “Isn’t that simply x² + 3²?”

Erklärus: “A very common mistake! (x+3)² means (x+3)×(x+3) – the whole bracket is taken twice, not just each number squared on its own. Let’s see it square by square, then it will be clear straight away – and perhaps the square search area will even help us narrow down your home.”

➕ First binomial formula – the plus formula

Imagine (a+b)² as the area of a square whose side is a+b long.

a² ab ab b² a b a b
Liri: “I can see four squares: one a², two ab squares and one b². Together: a² + ab + ab + b² = a² + 2ab + b²!”
Erklärus: “That is exactly the formula. The two ab squares are the reason why 2ab is in the middle – not just ab.”
1st binomial formula

(a+b)² = a² + 2ab + b²

Example: (x+3)² = x² + 2×x×3 + 3² = x² + 6x + 9

Solve: (x+5)² = x² + ?x + 25
Liri: “If I know the side length of my search area, I can use this formula to work out the whole area straight away – without measuring everything separately!”

➖ Second binomial formula – the minus formula

Erklärus: “With (a−b)² we start from the big square a² and cut off a strip of width b twice.”
(a−b)² −ab −ab +b²
Liri: “I cut off a strip at the top (−ab) and a strip on the right (−ab). But I have taken away the corner at the top right twice!”
Erklärus: “Well spotted – that is why you add this corner piece (b²) back once. a² − ab − ab + b² = a² − 2ab + b². What is left is exactly the golden square: (a−b)².”
2nd binomial formula

(a−b)² = a² − 2ab + b²

Example: (x−5)² = x² − 2×x×5 + 5² = x² − 10x + 25

Liri: “So the b² always stays plus, no matter whether there is a plus or a minus at the front?”
Erklärus: “Exactly – minus times minus makes plus. Only the middle term 2ab takes over the sign.”
Solve: (x−4)² = x² − ?x + 16

✖️ Third binomial formula – plus times minus (the difference of two squares)

Erklärus: “You can see this formula too. Take a square a² and cut out a small corner b² – what is left is an L-shape. This L-shape can be rearranged into a rectangle.”
a² minus b² (L-shape) b² a²−b² = Rectangle (a+b)×(a−b) a+b a−b
Liri: “So the L-shape and the rectangle have exactly the same area, just arranged differently?”
Erklärus: “Exactly – no piece is lost, none is added. Just rearranged. And that is exactly why: a² − b² = (a+b)×(a−b).”
🧠 Why does the middle term disappear?

(a+b)(a−b) = a² − ab + ab − b² = a² − b²

−ab and +ab are equal in size but opposite – they cancel each other out.

3rd binomial formula

(a+b)(a−b) = a² − b²

Example: (x+4)(x−4) = x² − 16

Solve: (x+6)(x−6) = x² − ?

🔄 Backwards – from the result to the bracket

The formulas also work backwards: from a² − b² you can straight away make (a+b)(a−b) again.

Erklärus: “If you see x² − 9, recognise straight away: 9 = 3², so that is (x+3)(x−3). That will help you enormously later when simplifying fractions.”
Liri: “For that I have to know the square numbers well – 1, 4, 9, 16, 25…”
Erklärus: “Exactly. Anyone who recognises 12² = 144 straight away finds the bracket in their head.”
Factorise: x² − 25 = (x+5)(x−?)
🌌 Binomial formulas in the universe

When astronomers work out the areas of sections of star maps that are made of combined squares, they use exactly this principle of splitting up.
Expressions such as (v+w)² also appear in physics when two speeds add up – the same formula, different letters.

❌ Common mistake

(a+b)² is not a² + b²! The middle term 2ab is often forgotten. Remember: (a+b)² = a² + 2ab + b² – never without the middle term.

📝 Exercise 1
Solve: (x+2)² = x² + 4x + ?
📝 Exercise 2
Solve: (x−7)² = x² − 14x + ?
📝 Exercise 3
Factorise: x² − 64 = (x+8)(x−?)

Three formulas, one picture in your head

Liri no longer had to learn three formulas by heart – she simply saw the divided square in front of her. And with the calculated area she could finally narrow down her search area.

Liri: “With the binomial formulas I can work out huge square search areas in a flash – or work back from a known area straight to the side lengths. That brings me a good deal closer to home.”
Erklärus: “That is exactly how real cartographers work: they split big, confusing areas into simple squares and rectangles. Anyone who can do that doesn’t get lost in complicated maps – and finds the way back more easily.”
Liri: “I don’t have to learn by heart if I understand why it is so.”
“(a+b)²=a²+2ab+b². (a−b)²=a²−2ab+b². (a+b)(a−b)=a²−b². Always take the whole bracket twice, never just square each part on its own.”
What you learned today
(a+b)²=a²+2ab+b²
Three binomial formulas – usable forwards and backwards
🌟

Chapter 46 mastered!

Liri can now confidently expand and factorise brackets in both directions.

Erklärus’s notebook – page 46:
“Remember the binomial formulas as a structure, not as letters: (First+Second)²=First²+2×First×Second+Second². That way you recognise them with other variables too.”
← Back to overview 46 – Binomial Formulas