Completing the Square
Rewrite a quadratic as a perfect square plus constant.
Key idea
Rewrite a quadratic as a perfect square plus constant.
In depth
Rewrite x² + bx as (x + b/2)² − (b/2)² to turn a quadratic into a perfect square plus a constant. It solves any quadratic and gives the vertex form of a parabola.
Worked example
x² + 6x + 5 = 0 → (x + 3)² − 9 + 5 = 0 → (x + 3)² = 4 → x = −1 or x = −5.
Formula
Real-world uses
- Finding the maximum height of a thrown object
- Deriving the quadratic formula
- Graphing parabolas
Common mistake
Adding (b/2)² to one side only. Balance it on the other side.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Completing the Square?
Rewrite a quadratic as a perfect square plus constant. It is a topic at the advanced level in Algebra.
What is the formula for Completing the Square?
The key formula is x²+bx = (x+b/2)² − b²/4.
Where is Completing the Square used in real life?
Real-world examples: Finding the maximum height of a thrown object; Deriving the quadratic formula; Graphing parabolas.
What should I learn before Completing the Square?
Make sure you are comfortable with Quadratic Equations by Factoring, the previous topic in Algebra.
What is a common mistake with Completing the Square?
Adding (b/2)² to one side only. Balance it on the other side.
How can I practise Completing the Square?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Completing the Square?
Move on to Quadratic Formula next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Quadratic Formula – Gives roots of any quadratic.