Completing the Square: worked example
| Coefficient a | 1 |
|---|---|
| Coefficient b | 6 |
| Constant c | 5 |
| Vertex x (h) | -3 |
| Vertex y (k) | -4 |
Completing the Square: formula
Enter the coefficients a, b and c of the quadratic ax² + bx + c. The calculator rewrites it as a times x minus h squared plus k, where h equals minus b over 2a and k equals c minus b squared over 4a. The coefficient a cannot be zero.
If a is not 1, the method factors it out first, so check the signs of b and c.
Completing the Square: changing Coefficient a
| Coefficient a | Vertex x (h) | Vertex y (k) |
|---|---|---|
| -1 | 3 | 14 |
| 1 (example) | -3 | -4 |
| 2 | -1.5 | 0.5 |
| 3 | -1 | 2 |
Completing the Square: compare input ranges
| Input | Tested from | To | Vertex x (h): low input | Vertex x (h): high input | Swing |
|---|---|---|---|---|---|
| Coefficient b | 4 | 8 | -2 | -4 | 2 (67%) |
| Constant c | 4 | 6 | -3 | -3 | 0 |
Completing the Square: individual input comparisons
| Input | Value | Vertex x (h) | Vertex y (k) |
|---|---|---|---|
| Coefficient a | -1 | 3 | 14 |
| 1 | -3 | -4 | |
| 2 | -1.5 | 0.5 | |
| 3 | -1 | 2 | |
| Coefficient b | 2 | -1 | 4 |
| 4 | -2 | 1 | |
| 6 | -3 | -4 | |
| 8 | -4 | -11 | |
| 10 | -5 | -20 | |
| Constant c | 3 | -3 | -6 |
| 4 | -3 | -5 | |
| 5 | -3 | -4 | |
| 6 | -3 | -3 | |
| 7 | -3 | -2 |
Completing the Square: combined input changes
| Coefficient a / Coefficient b | 2 | 6 | 10 |
|---|---|---|---|
| -1 | 1 | 3 | 5 |
| 0 | Undefined for these inputs | Undefined for these inputs | Undefined for these inputs |
| 1 | -1 | -3 | -5 |
| 2 | -0.5 | -1.5 | -2.5 |
| 3 | -0.333 | -1 | -1.7 |
Completing the Square: which assumptions move the result?
Completing the Square: values and definitions
| Quantity and definition | Value used |
|---|---|
| Coefficient a — It cannot be zero. | 1 |
| Coefficient b | 6 |
| Constant c | 5 |
| Vertex x (h) | -3 |
| Vertex y (k) | -4 |
Common error: Reading the vertex x as +3. The form (x + 3) squared has its turning point where the bracket is zero, which is at x = -3.
The method gives the same roots as the quadratic formula but through a different route. It requires a nonzero a, and with a negative a the vertex is a maximum. Fractions can appear in h and k, so exact values are better kept as fractions.
Complex roots appear when the vertex is above the axis for an upward parabola.
Algebra & Arithmetic calculators: Dot Product Calculator, Mixed Number to Improper Fraction Calculator, Exponential Growth Calculator, Quadratic Formula Calculator, Fraction and Ratio Simplifier Calculator, Exponent Calculator.
Completing the square rewrites a quadratic so its turning point shows on the surface. For x squared plus 6x plus 5, the form is (x + 3) squared minus 4, so the lowest point sits at x = -3 with a value of -4. That vertex is what you need to find the maximum area, the minimum cost or the peak of a thrown ball without graphing anything.
Confirm the coefficient a is not zero. Without an x squared term the curve is a straight line and has no vertex.
Solve for roots on the quadratic formula calculator, compute distance and slope on the distance and slope page, and see the exponent and root calculators for arithmetic support. The exponential growth page shows another model, and the arc and ellipse calculators cover curved shapes.
What is completing the square used for?
To rewrite a quadratic in vertex form, find its maximum or minimum, solve equations and derive the quadratic formula.
How do I find the vertex of a parabola?
The x coordinate is minus b over 2a, and the y coordinate is found by substituting it into the equation.
What does the sign of a tell me?
If a is positive the parabola opens upward and the vertex is a minimum, and if negative it opens downward and the vertex is a maximum.