Completing the Square
Completing the square rewrites a quadratic as a squared bracket plus a constant, revealing the turning point.
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Completing the square rewrites a quadratic as a squared bracket plus a constant, revealing the turning point.
- x squared + bx becomes (x + b/2) squared minus (b/2) squared
- The turning point is at x equals -b/2
- It can be used to solve quadratics
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Completing the Square?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Completing the Square. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Completing the Square in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Completing the Square gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Completing the Square in detail
Completing the square rewrites a quadratic as a(x + p)² + q. For x² + 6x + 5, halve the coefficient of x to get 3, so the expression becomes (x + 3)² − 9 + 5 = (x + 3)² − 4.
This form reveals the turning point of the parabola at (−p, q), here (−3, −4), and shows the minimum or maximum value. It can also be used to solve quadratics that do not factorise, by rearranging and taking square roots, and it is the method behind deriving the quadratic formula.
How the interactive helps you understand Completing the Square
Move the sliders to change the values in the equation. The curve, the moving point and, where they exist, the roots (the points where the curve crosses the x-axis) update straight away.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Completing the Square.
Real-life uses and examples
- Check a homework answer about Completing the Square by estimating first, then working it out step by step.
- Use Completing the Square when comparing prices, discounts or recipe quantities in everyday life.
- Explain Completing the Square to a friend with a simple picture, such as a row of tiles or a bag of sweets.
Common mistakes to avoid
- Memorising the steps for Completing the Square without checking why they work.
- Forgetting units, signs or brackets when you apply Completing the Square.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Completing the Square
What is Completing the Square?
Completing the square rewrites a quadratic as a squared bracket plus a constant, revealing the turning point.
What are the most important points about Completing the Square?
The main points are: x squared + bx becomes (x + b/2) squared minus (b/2) squared; The turning point is at x equals -b/2; It can be used to solve quadratics.
Why is Completing the Square useful?
Understanding Completing the Square gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Completing the Square?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Completing the Square?
Related topics in Mathematics include Coordinates and Graphs, Midpoint and Distance, Exponential Graphs.