Completing the Square Explained Step by Step

Completing the square rewrites a quadratic so one side is a perfect square like (x + 3)². Once it is in that shape, you can solve by taking a square root. It is also the method behind vertex form and the quadratic formula itself.

Die Idee in Kürze

For x² + bx, adding (b/2)² makes a perfect square: x² + bx + (b/2)² = (x + b/2)². Add the same amount to both sides to keep the equation balanced.

Schritte

1. Move the constant term to the right side. If a is not 1, divide every term by a first.
2. Take half of b and square it.
3. Add that number to both sides, then write the left side as a squared binomial.
4. Take the square root of both sides (remember ±) and solve for x.

Beispielaufgabe

Solve x² + 6x − 7 = 0 by completing the square.

Vollständige Lösung

Schritt für Schritt erklärt

Schritt 1

Add 7 to both sides so the x terms are alone on the left.

Schritt 2

Half of 6 is 3, and 3² is 9. Add 9 to both sides: the right side becomes 16.

Schritt 3

The left side is now the perfect square (x + 3)².

Schritt 4

Square root both sides: x + 3 = ±4, so x = 1 or x = −7.

Häufiger Fehler

Add (b/2)² to both sides, not just the left. Adding it to one side only changes the equation and gives the wrong answers.

Übungsaufgaben

1. Solve x² + 4x − 5 = 0 by completing the square.
2. Solve x² − 8x + 12 = 0 by completing the square.
3. Solve x² + 2x − 2 = 0 by completing the square.

Lösungen

1. x = 1 or x = −5
2. x = 6 or x = 2
3. x = −1 ± √3

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Passende Guides

• Die Mitternachtsformel (abc-Formel) erklärt
• Vertex Form and How to Graph a Parabola
• Quadratische Terme faktorisieren