Answer first
What this calculator tells you
Rewrite a quadratic in vertex form and find the vertex from its three coefficients. Find the highest or lowest point of a parabola without graphing it. Formula: a x² + b x + c = a (x − h)² + k, with h = −b ÷ (2a) and k = c − b² ÷ (4a). At the worked-example inputs, the vertex x (h) is -3. Holding every other input steady, moving coefficient a from -1 to 3 moves the result from -3 to 3.
Transparent method
The formula
Find the highest or lowest point of a parabola without graphing it.
Worked example
Example inputs
How to interpret the result
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.
At the worked-example inputs the vertex x (h) is -3. It falls as coefficient b increases; constant c does not move it.
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects.
Before you rely on it
What to check
Confirm the coefficient a is not zero. Without an x squared term the curve is a straight line and has no vertex.
The common error
Where people go wrong with completing the square calculator
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.
Sensitivity evidence
How coefficient a changes the vertex x (h)
Holding every other input at the worked-example value, moving coefficient a from -1 to 3 moves the vertex x (h) from -3 to 3: a spread of 6, or 200% of the worked-example result.
| Coefficient a | Vertex x (h) | Vertex y (k) |
|---|---|---|
| -1 | 3 | 14 |
| 0 | : | : |
| 1worked example | -3 | -4 |
| 2 | -1.5 | 0.5 |
| 3 | -1 | 2 |
Every input, tested
Which input moves the vertex x (h) most
Of the 2 inputs, only coefficient b moves the vertex x (h): moving coefficient b from 4 to 8 takes the vertex x (h) from -2 to -4, a swing of 67% of the worked-example figure. Constant c does not change it at all.
| Input | Tested from | To | Vertex x (h) at each end | Swing |
|---|---|---|---|---|
| Coefficient b | 4 | 8 | -2 to -4 | 2 (67%) |
| Constant c | 4 | 6 | -3 to -3 | none |
Two variables at once
Vertex x (h) by coefficient a and coefficient b
Across the grid the vertex x (h) runs from -5 to 5. Moving coefficient a from -1 to 3 shifts it by 4 at the middle column, and moving coefficient b from 2 to 10 shifts it by 4 at the middle row, so neither is the bigger lever here.
| Coefficient a \ Coefficient b | 2 | 6 | 10 |
|---|---|---|---|
| -1 | 1 | 3 | 5 |
| 0 | : | : | : |
| 1 | -1 | -3 | -5 |
| 2 | -0.5 | -1.5 | -2.5 |
| 3 | -0.333 | -1 | -1.7 |
The highlighted cell is the worked example: -3.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Coefficient a | 1 | It cannot be zero. |
| Coefficient b | 6 | Enter the coefficient b used in this calculation. |
| Constant c | 5 | Enter the constant c used in this calculation. |
| Vertex x (h) | -3 | |
| Vertex y (k) | -4 | |
Inputs, definitions and assumptions
Coefficient a
It cannot be zero. The prefilled worked-example value is 1.
Coefficient b
Enter the coefficient b used in this calculation. The prefilled worked-example value is 6.
Constant c
Enter the constant c used in this calculation. The prefilled worked-example value is 5.
How to use this calculator
- 1Verify the inputs. Gather coefficient a, coefficient b and constant c from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the vertex x (h) at -3. Store your own version of it as Scenario A.
- 3Test one change. Start with coefficient b, the input with the biggest effect here: moving coefficient b from 4 to 8 takes the vertex x (h) from -2 to -4, a swing of 67% of the worked-example figure.
- 4Check the boundary. Read the interpretation boundary above before acting on the result.
People also ask
Frequently asked questions
How do you calculate completing the square?
a x² + b x + c = a (x − h)² + k, with h = −b ÷ (2a) and k = c − b² ÷ (4a). At the worked-example inputs the vertex x (h) is -3.
What does the completing the square result mean?
Find the highest or lowest point of a parabola without graphing it. At the worked-example inputs the vertex x (h) is -3. It falls as coefficient b increases; constant c does not move it.
How much does coefficient a change the vertex x (h)?
Holding every other input at the worked-example value, moving coefficient a from -1 to 3 moves the vertex x (h) from -3 to 3, a spread of 6.
What are the limits of this completing the square calculator?
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects. The tables on this page test coefficient a only from -1 to 3; a value outside that range is not tabulated here.
How much does coefficient b matter in the completing the square calculator?
The worked example uses 6. With the other inputs left at the worked example, moving coefficient b from 4 to 8 takes the vertex x (h) from -2 to -4, a swing of 67% of the worked-example figure.
How much does constant c matter in the completing the square calculator?
The worked example uses 5. The vertex x (h) does not depend on constant c; it moves the vertex y (k) from -5 to -3 instead when constant c goes from 4 to 6.
Which inputs change the vertex y (k) in the completing the square calculator?
At the worked-example inputs it is -4. Coefficient b takes it from 1 to -11 and constant c takes it from -5 to -3.
Why does a quadratic equation sometimes have no real solution?
When the discriminant (b²−4ac) is negative, the parabola described by the equation never crosses the x-axis. So there is no real value of x that makes it zero. The equation still has two solutions. But they are complex numbers: outside what a real-number calculator displays.
How do Roman numerals show 4 and 9?
By subtraction: a smaller symbol before a larger one is taken away. IV is 4, IX is 9, XL is 40 and XC is 90. The same pattern gives CD for 400 and CM for 900.
Sources and evidence
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