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Zeros and the pq formula

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

A zero (die Nullstelle) is a point on the \(x\)-axis. You calculate it by taking roots and with the pq formula, and you know when there is none.
📄 Open worksheet (PDF, German)

The video

In the video you see the zeros as points on the \(x\)-axis. The calculation stands next to it, step by step.

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

Contents of the video

  1. 0:00 Zeros
  2. 0:30 Taking the root
  3. 0:53 With the vertex form
  4. 1:20 The pq formula
  5. 1:47 Calculating with the pq formula
  6. 2:09 The discriminant
  7. 2:45 Careful, mistake!
  8. 3:10 Try it yourself

The most important points

Zeros

A zero is a point on the \(x\)-axis: there \(y = 0\). A parabola has two, one or no zero.

Two, one and no zero.
Two, one and no zero.

Taking the root

\(x^2 - 4 = 0\) \(\to\) \(x^2 = 4\) \(\to\) \(x = 2\) or \(x = -2\). In vertex form: \((x - 1)^2 - 4 = 0\) \(\to\) \(x - 1 = \pm 2\) \(\to\) \(x = 3\) or \(x = -1\).

x^2 - 4 = 0 has the zeros -2 and 2.
\(x^2 - 4 = 0\) has the zeros \(-2\) and \(2\).
(x - 1)^2 - 4 = 0 has the zeros -1 and 3.
\((x - 1)^2 - 4 = 0\) has the zeros \(-1\) and \(3\).

pq formula

\(x^2 + px + q = 0\) has the solutions \(x = -\frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q}\). Example \(x^2 - 2x - 3 = 0\): \(p = -2\), \(q = -3\); \(-\frac{p}{2} = 1\); \(\sqrt{1 + 3} = 2\). So \(x = 3\) or \(x = -1\).

Discriminant (die Diskriminante) \(\left(\frac{p}{2}\right)^2 - q\): positive \(\to\) two zeros, zero \(\to\) one, negative \(\to\) none. Example \(x^2 + 2x + 5\): \(1 - 5 = -4\), so no zero. Careful: For \(p = -2\) we get \(-\frac{p}{2} = +1\), not \(-1\).

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

Calculate the zeros of \(y = x^2 + 2x - 8\).

Show solution

\(p = 2\), \(q = -8\): \(x = -1 \pm \sqrt{1 + 8} = -1 \pm 3\), so \(x = 2\) or \(x = -4\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Nullstellen quadratischer Funktionen (Wurzelziehen, pq-Formel, Diskriminante). 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.