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.

L'idée en bref

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.

Les étapes

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.

Exemple résolu

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

Solution complète

Explication étape par étape

Étape 1

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

Étape 2

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

Étape 3

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

Étape 4

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

Erreur fréquente

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.

Exercices d'entraînement

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.

Corrigés

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

Demandez à PennPaper de vous l'expliquer en direct

Si les étapes vous semblent claires mais que vous bloquez encore, lancez une séance PennPaper et demandez au tuteur de dessiner le problème sur le tableau blanc. Voir les symboles bouger un à un, c'est souvent ce qui fait tilt.

Guides associés

• La formule des racines (discriminant) expliquée
• Vertex Form and How to Graph a Parabola
• Comment factoriser un trinôme du second degré