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Binomial formulas

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

A square has side 5. What happens to the area (die Fläche) when the side gets 2 longer or shorter? From the picture we get the three binomial formulas (die binomischen Formeln).
📄 Open worksheet (PDF, German)

The video

Watch the colours: \( a \) and \( a^2 \) are blue, \( b \) and \( b^2 \) are purple, the strips \( ab \) are orange.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:24 Plus two
  3. 1:03 The first binomial formula
  4. 1:30 Examples
  5. 1:58 Minus two
  6. 2:38 The second binomial formula
  7. 3:03 Longer and shorter
  8. 3:43 The third binomial formula
  9. 4:11 All three formulas
  10. 4:30 Try it yourself

The most important points

Side 5: area \( 25 \). Side \( 5 + 2 = 7 \): area \( 7^2 = 49 \) – not \( 25 + 4 = 29 \)! The picture shows why: \( 25 + 10 + 10 + 4 = 49 \). The old square, two strips and a small square in the corner.

The old square (25), two strips (10 each) and the corner (4).
The old square (25), two strips (10 each) and the corner (4).

1st binomial formula

\( (a + b)^2 = a^2 + 2ab + b^2 \). Don't forget the two strips! Examples: \( 11^2 = (10 + 1)^2 = 100 + 20 + 1 = 121 \), \( (x + 3)^2 = x^2 + 6x + 9 \).

Side \( 5 - 2 = 3 \): area \( 9 = 25 - 10 - 10 + 4 \). The corner was cut off twice, so it comes back once.

Two strips off, the corner back once: 25 - 10 - 10 + 4 = 9 .
Two strips off, the corner back once: \( 25 - 10 - 10 + 4 = 9 \).

2nd binomial formula

\( (a - b)^2 = a^2 - 2ab + b^2 \). At the end there is plus \( b^2 \). Example: \( 9^2 = (10 - 1)^2 = 100 - 20 + 1 = 81 \).

One side +2 , the other -2 : cut out the corner, turn the piece – the rectangle 7 · 3 = 21 = 25 - 4 .
One side \( +2 \), the other \( -2 \): cut out the corner, turn the piece – the rectangle \( 7 \cdot 3 = 21 = 25 - 4 \).

3rd binomial formula

\( (a + b) \cdot (a - b) = a^2 - b^2 \). Examples: \( 21 \cdot 19 = (20 + 1)(20 - 1) = 400 - 1 = 399 \), \( (x + 5)(x - 5) = x^2 - 25 \).

Careful, common mistake: \( (a + b)^2 \) is not \( a^2 + b^2 \).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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The worksheet – Page 5
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Show the worksheet solutions
The worksheet – Page 6
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Try it yourself

Task

Calculate with the binomial formulas: a) \( 31^2 \) b) \( 19^2 \) c) \( 12 \cdot 8 \) d) \( (x + 4)^2 \)

Show solution

a) \( 900 + 60 + 1 = 961 \) b) \( 400 - 40 + 1 = 361 \) c) \( 100 - 4 = 96 \) d) \( x^2 + 8x + 16 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: the three binomial formulas (a + b)², (a − b)² and (a + b)(a − b), also for mental arithmetic. Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.