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Lesson 3 of 5

Completing the Square

Completing the square is a technique for rewriting a quadratic in vertex form. It always works — even when factoring fails.

Core Concept

Vertex Form

We want to rewrite ax² + bx + c as a(x − h)² + k. This is called vertex form, and it reveals the vertex of the parabola at (h, k).

Method (a = 1)

Step-by-Step: Completing the Square

Solve x² + 6x + 5 = 0 by completing the square.

1. Move constant: x² + 6x = −5
2. Take half of b, then square it: (6/2)² = 9
3. Add 9 to BOTH sides: x² + 6x + 9 = −5 + 9
4. Left side is now a perfect square: (x + 3)² = 4
5. Take square root: x + 3 = ±2
6. x = −3 + 2 = −1 or x = −3 − 2 = −5

Example 2

Rewriting in Vertex Form

Rewrite f(x) = x² − 4x + 7 in vertex form.

1. Group x terms: f(x) = (x² − 4x) + 7
2. Half of −4 is −2; (−2)² = 4
3. Add and subtract 4 inside: (x² − 4x + 4) − 4 + 7
4. f(x) = (x − 2)² + 3

Vertex: (2, 3). The parabola has its minimum at x = 2, y = 3.

Watch Out

Common Mistakes

Your Turn

Try It Yourself

Q: Solve x² − 8x + 7 = 0 by completing the square.

Show Answer
x² − 8x = −7
Add (−8/2)² = 16: x² − 8x + 16 = 9
(x − 4)² = 9
x − 4 = ±3
x = 7 or x = 1

Key Takeaways

1-Minute Summary

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