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Vieta's theorem

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

For \(x^2 + px + q = 0\) the solutions are closely linked to \(p\) and \(q\). So you can guess solutions and set up equations.
📄 Open worksheet (PDF, German)

The video

In the video we discover the theorem with \(x^2 - 5x + 6 = 0\) and guess the solutions of \(x^2 + x - 12 = 0\).

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

Contents of the video

  1. 0:00 Introduction
  2. 0:21 Vieta's theorem
  3. 0:51 Guessing solutions
  4. 1:20 Setting up an equation
  5. 1:43 Careful, mistake!
  6. 2:09 Try it yourself

The most important points

Vieta's theorem

If \(x^2 + px + q = 0\) has the solutions \(x_1\) and \(x_2\), then: \(x_1 + x_2 = -p\) and \(x_1 \cdot x_2 = q\) (der Satz von Vieta).

Reason: (x - x_1)(x - x_2) = x^2 - (x_1 + x_2)· x + x_1 x_2.
Reason: \((x - x_1)(x - x_2) = x^2 - (x_1 + x_2)\cdot x + x_1 x_2\).

Guessing: \(x^2 + x - 12 = 0\). We need two numbers with product \(-12\) and sum \(-1\): \(-4\) and \(3\).

Guessing solutions.
Guessing solutions.

The other way round: solutions \(-2\) and \(5\) give \(p = -3\), \(q = -10\), so \(x^2 - 3x - 10 = 0\).

Setting up an equation.
Setting up an equation.

Careful: the sum is \(-p\), not \(p\). For \(x^2 - 5x + 6 = 0\), \(x_1 + x_2 = 5\).

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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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

Guess the solutions of \(x^2 - 7x + 10 = 0\) with Vieta's theorem.

Show solution

\(2 \cdot 5 = 10\) and \(2 + 5 = 7\): \(x_1 = 2\), \(x_2 = 5\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Satz von Vieta (x₁ + x₂ = −p, x₁ · x₂ = q, Lösungen raten, Gleichung aufstellen). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

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.