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Standard form and completing the square

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

A parabola can be written in two ways: standard form and vertex form. You learn how to get from one to the other.
📄 Open worksheet (PDF, German)

The video

In the video the terms are rewritten line by line. At the end the graph shows the vertex.

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

Contents of the video

  1. 0:00 Two forms
  2. 0:27 Expanding the bracket
  3. 0:54 Completing the square
  4. 1:21 The vertex
  5. 1:48 Careful, mistake!
  6. 2:22 Try it yourself

The most important points

Vertex form → standard form

Expand the bracket with the binomial formula (die binomische Formel): \((x - 2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5\).

Expanding the bracket.
Expanding the bracket.

Standard form → vertex form

Completing the square (die quadratische Ergänzung) for \(y = x^2 + 6x + 5\): Halve the number in front of \(x\): \(6 : 2 = 3\). Square it: \(3^2 = 9\). Add and subtract the 9: \(x^2 + 6x + 9 - 9 + 5\).

Adding and subtracting the 9.
Adding and subtracting the 9.

The first three terms are \((x + 3)^2\). What is left is \(-9 + 5 = -4\). So \(y = (x + 3)^2 - 4\) and \(S(-3|-4)\).

The parabola y = x^2 + 6x + 5 with S(-3|-4).
The parabola \(y = x^2 + 6x + 5\) with \(S(-3|-4)\).

Careful: Writing only \(+9\) is wrong. You must write \(-9\) too. Check with \(x = 0\): standard form \(y = 5\); correct \((0 + 3)^2 - 4 = 5\).

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
Page 1/3
The worksheet – Page 2
Page 2/3
Show the worksheet solutions
The worksheet – Page 3
Page 3/3

Try it yourself

Task

Write \(y = x^2 - 8x + 10\) in vertex form. What is the vertex?

Show solution

\(y = (x - 4)^2 - 16 + 10 = (x - 4)^2 - 6\). \(S(4|-6)\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Normalform und Scheitelpunktform, quadratische Ergänzung. 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.