Completing the Square
Add (b/2)² to turn a quadratic into a perfect square.
About this topic
Completing the square rewrites x² + bx + c as a perfect square plus a constant. Add (b/2)² to both sides, so x² + 6x becomes x² + 6x + 9 = (x + 3)². This makes any quadratic solvable with the square root method, even one that will not factor.
It is also how you convert a quadratic from standard form to vertex form. To solve x² + 6x − 7 = 0: move the constant to get x² + 6x = 7, add 9 to both sides to get (x + 3)² = 16, then x + 3 = ±4, giving x = 1 or x = −7. If the leading coefficient is not 1, divide through by it first. A geometric picture is a square with a missing corner that you complete by adding a small square.
The technique matters because it proves the quadratic formula and also appears in the equations of circles and in calculus integrals. It reveals the vertex, so you can find maximum area or profit.
Remember to add the same amount to both sides, to halve b before squaring it, and to handle a fractional middle coefficient carefully. Forgetting to balance the equation is the most frequent error.
Key ideas
- Move the constant over
- Add (b/2)² to both sides
- Write as (x + b/2)²
Quick check
Use cases: where Completing the Square helps
- Homework: review the key idea behind Completing the Square before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Completing the Square without notes.
- Visual learning: play with the interactive lab to see how Completing the Square changes when you adjust its inputs.
- Teaching a friend: explain Completing the Square in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Completing the Square?
Add (b/2)² to turn a quadratic into a perfect square.
What will I learn in Completing the Square?
Move the constant over. Add (b/2)² to both sides. Write as (x + b/2)².
Is Completing the Square part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Completing the Square lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Completing the Square?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.