Calculator
What this solves
Enter coefficients for ax² + bx + c. The calculator rewrites it into completed square form, shows the vertex, and reports real roots when they exist.
Input tips
- Use decimals or scientific notation like 1.2e-3.
- If a = 0, the expression is linear or constant.
- Turn on fraction mode to see cleaner symbolic terms.
Example data
| Quadratic | Completed square form | Vertex |
|---|---|---|
| x² + 6x + 5 | (x + 3)² − 4 | (-3, -4) |
| 2x² − 8x + 5 | 2(x − 2)² − 3 | (2, -3) |
| -3x² + 12x − 1 | -3(x − 2)² + 11 | (2, 11) |
These examples help you verify outputs and learn the pattern.
Formula used
For ax² + bx + c with a ≠ 0:
The vertex is (h, k) where h = −b/2a and k = f(h). This form makes the turning point clear.
The discriminant is D = b² − 4ac. If D ≥ 0 there are real roots; otherwise roots are complex.
How to use this calculator
- Enter a, b, and c for your quadratic.
- Select your preferred decimal precision for the displayed results.
- Enable fraction mode if you want cleaner symbolic values.
- Click Submit to see the completed square form and steps.
- Use the download buttons to export CSV or PDF results.
FAQs
1) What does “completing the square” achieve?
It rewrites a quadratic into a perfect-square expression plus a constant. This reveals the vertex quickly and makes graphing and solving easier.
2) Can I use this when a equals zero?
Yes, but it becomes a linear or constant expression. The calculator switches to a simple solve and notes that completing the square is not applicable.
3) Why does the vertex show up naturally?
In the form a(x − h)² + k, the squared term is minimized at x = h. The value there is k, giving the vertex.
4) What if the discriminant is negative?
A negative discriminant means no real roots. The quadratic still has a real vertex and completed square form, but the x-intercepts are complex.
5) When should I enable fraction mode?
Use it for integer coefficients or teaching steps. It helps show exact values like b/2a and c − b²/4a without rounding noise.
6) Does rounding affect correctness?
Rounding affects display only. Internally the calculator uses floating-point math, and then formats results to your selected precision for readability.
7) How do I verify my result manually?
Expand the completed square form to recover ax² + bx + c. Matching coefficients confirms the transformation is correct.