Completing the square
Rewrite as .
Method. Convert to .
Steps.
- Factor from the first two terms: .
- Half the coefficient of , square it: , then .
- Add and subtract inside the bracket: .
- Simplify: .
Worked example. .
- Factor 2 from first two: .
- Half of is ; squared is : .
- Expand: .
Why this is useful.
- Vertex. Minimum (if ) at , . So has min at .
- Range. for ; for .
- Sketching. Tells you vertex and shape.
Cambridge tip. Mark schemes often reward "completed square form" PLUS extracting vertex and minimum from it.
- Factor from first two terms.
- Half-square-the-x-coefficient.
- Vertex at , .