Calculator Inputs
Example Data Table
| Root 1 | Root 2 | a | Derived quadratic equation |
|---|---|---|---|
| 2 | 5 | 1 | x2 - 7x + 10 = 0 |
| -3 | 4 | 1 | x2 - x - 12 = 0 |
| 1.5 | -2 | 2 | 2x2 + x - 6 = 0 |
| 3 | 3 | 1 | x2 - 6x + 9 = 0 |
Formula Used
A quadratic equation built from two roots uses the factored model a(x - r1)(x - r2) = 0. Here, r1 and r2 are the given roots, and a is the leading coefficient.
a(x - r1)(x - r2) = ax2 - a(r1 + r2)x + a(r1r2)
After expansion, the standard coefficients become:
c = a(r1r2)
The axis of symmetry is the midpoint of the roots: x = (r1 + r2) / 2. The discriminant is b2 - 4ac. These values help describe the graph and confirm root behaviour.
How to Use This Calculator
Enter the first root in the Root 1 field.
Enter the second root in the Root 2 field.
Set the leading coefficient if you need scaling.
Pick a decimal precision for displayed results.
Optionally set graph minimum and maximum x values.
Click Derive Equation to generate the quadratic.
Review the standard, factored, and vertex forms.
Check coefficients, discriminant, intercepts, and graph direction.
Use the export buttons to save CSV or PDF output.
Frequently Asked Questions
1. How is the quadratic equation created from two roots?
Use y = a(x - r₁)(x - r₂). Expand the brackets to get ax² - a(r₁ + r₂)x + ar₁r₂. Set that expression equal to zero for the quadratic equation.
2. Why does the calculator ask for a leading coefficient?
The leading coefficient scales the parabola. It changes the steepness, vertex height, and coefficients, while keeping the same roots unless the coefficient is zero.
3. What happens when both roots are the same?
The equation has a repeated root. Its factored form becomes a(x - r)², and the discriminant becomes zero. The graph touches the x-axis once.
4. How are b and c found quickly?
For roots r₁ and r₂, coefficient b equals -a(r₁ + r₂). Constant c equals ar₁r₂. These come directly from expanding the factored form.
5. What does the discriminant tell me here?
The discriminant shows root behaviour. A positive value means two real roots. Zero means one repeated real root. A negative value means complex roots.
6. Why is the axis of symmetry the average of the roots?
A parabola is symmetric about its centre line. The midpoint between the two roots is that centre, so x = (r₁ + r₂) / 2 gives the axis.
7. Can I use decimal roots?
Yes. The calculator accepts integers and decimals. It then expands the quadratic and rounds displayed values to your chosen precision for cleaner output.
8. What do the CSV and PDF downloads include?
They include the derived equation, coefficient values, forms, intercept details, discriminant, symmetry line, and graph-related summary values from the current calculation.