targas-edu.de
DE ☾ A+

Completing the square with a factor a

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

If there is a number \(a \neq 1\) in front of \(x^2\), you factor it out first. Then you complete the square as usual and find the vertex.
📄 Open worksheet (PDF, German)

The video

In the video we bring \(y = 2x^2 - 8x + 5\) and \(y = -x^2 + 6x - 5\) into vertex form.

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 Factoring out a
  3. 0:39 Completing and simplifying
  4. 1:22 Negative a
  5. 1:55 Careful, mistake!
  6. 2:23 Try it yourself

The most important points

The steps

\(y = 2x^2 - 8x + 5\). 1. Factor out only from the terms with \(x\): \(2\cdot(x^2 - 4x) + 5\). 2. Complete (quadratische Ergänzung): \(2\cdot(x^2 - 4x + 4 - 4) + 5\). 3. Simplify: \(2\cdot(x-2)^2 - 8 + 5 = 2\cdot(x-2)^2 - 3\). So \(S(2\,|\,-3)\).

The transformation step by step and the graph.
The transformation step by step and the graph.

Negative \(a\): \(y = -x^2 + 6x - 5 = -(x^2 - 6x) - 5 = -(x-3)^2 + 4\), so \(S(3\,|\,4)\), opens downwards.

Factoring out a = -1.
Factoring out \(a = -1\).

Careful: You must multiply the subtracted number by \(a\). Check with \(x = 0\): normal form and vertex form give the same value.

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 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

Bring \(y = 0{,}5x^2 + 2x - 1\) into vertex form.

Show solution

\(y = 0{,}5\cdot(x^2 + 4x) - 1 = 0{,}5\cdot(x+2)^2 - 3\), \(S(-2\,|\,-3)\). Check \(x = 0\): \(0{,}5\cdot 4 - 3 = -1\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: quadratische Ergänzung mit Faktor a ≠ 1 (ausklammern, ergänzen, Scheitelpunkt). 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.