Completing the Square Calculator
Calculate completing the square instantly in your browser.
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What this tool does
Calculate completing the square instantly in your browser. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
How to use the Completing the Square Calculator
- Enter or select coefficient b (x² + bx + c).
- Enter or select constant c.
- Read the calculated result; change any measurement to compare alternatives.
Formula
Take half of b, square it, and rewrite x² + bx + c as (x + b/2)² + (c − (b/2)²) — vertex form — which exposes the vertex and makes the equation solvable by taking square roots.
- b
- Coefficient b (x² + bx + c)
- c
- Constant c
Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
Worked example
For completing the square calculator, the following measurements illustrate the exact method: Coefficient b (x² + bx + c): 6; Constant c: 5.
Inputs
- Coefficient b (x² + bx + c)6
- Constant c5
Result
- Vertex form(x + 3)² − 4
- Vertex x (−b/2)-3
- Vertex y-4
- Half of b (b/2)3
- (b/2)²9
- Roots1 and -7
Results explained
- Vertex form
- Vertex form from the formula above. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
- Vertex x (−b/2)
- Vertex x (−b/2) from the formula above. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
- Vertex y
- Vertex y from the formula above. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
- Half of b (b/2)
- Half of b (b/2) from the formula above. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
- (b/2)²
- (b/2)² from the formula above. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
- Roots
- Roots from the formula above. Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
Frequently asked questions
Take half of b, square it, and rewrite x² + bx + c as (x + b/2)² + (c − (b/2)²) — vertex form — which exposes the vertex and makes the equation solvable by taking square roots.
Written for a leading coefficient of 1; if your x² coefficient is not 1, divide every term by it first.
Where mathematically meaningful, yes. Fields specify permitted ranges and undefined operations display an error.
Exact integer calculations use safe integers; decimal calculations use browser floating point, so small rounding differences can occur.
All values remain in your browser and are never uploaded.