Result
Enter coefficients and calculate to see the roots and their difference.
Formula Used
Quadratic equation: ax² + bx + c = 0
Discriminant: D = b² − 4ac
Roots: x₁,₂ = (−b ± √D) / (2a)
Signed difference: x₁ − x₂ = √D / a
Absolute real-root difference: |x₁ − x₂| = √D / |a|
When D is negative, the calculator reports complex conjugate roots.
How to Use This Calculator
- Enter coefficients a, b, and c.
- Select the desired decimal precision.
- Choose your preferred result display.
- Press Calculate to analyze the quadratic.
- Export the result as CSV or PDF.
About the Difference Between Quadratic Roots
A quadratic equation has the form ax² + bx + c = 0. Its roots are values that make the expression equal zero. The difference between those roots measures how far apart they are on the number line. This calculator finds both roots, the discriminant, the signed difference, and the absolute difference.
Why the Discriminant Matters
The discriminant is D = b² − 4ac. A positive discriminant gives two distinct real roots. A zero discriminant gives one repeated real root. A negative discriminant gives two complex conjugate roots. These cases control how the root difference should be interpreted.
Using the Direct Difference Formula
The quadratic formula normally calculates each root separately. However, subtracting the two formulas gives x₁ − x₂ = √D / a. For real roots, the nonnegative distance is √D / |a|. This direct expression is useful when only root separation matters.
Signed and Absolute Differences
The signed difference depends on the root ordering. This calculator defines x₁ using the plus sign and x₂ using the minus sign. Changing the order reverses the sign. The absolute difference removes ordering and gives the numerical spacing between real roots.
Exact and Decimal Results
Some quadratics have irrational roots. In those cases, exact radical notation can be more informative than decimals. Decimal approximations remain useful for measurement, plotting, and comparison. The calculator displays both when possible.
Useful Relationships
The sum of the roots is −b/a. Their product is c/a. These Vieta relationships help verify calculated roots. Together with the discriminant, they provide a compact summary of the equation. Complex roots are also handled clearly, including their imaginary difference.
Practical Use
Root separation appears in algebra, graph analysis, optimization, and equation checking. Larger differences indicate roots farther apart. A zero difference means the parabola touches the axis once. Use exported results for assignments, records, or further analysis.
Example Table
| a | b | c | Discriminant | Root 1 | Root 2 | Absolute Difference |
|---|---|---|---|---|---|---|
| 1 | -5 | 6 | 1 | 3 | 2 | 1 |
| 2 | -7 | 3 | 25 | 3 | 0.5 | 2.5 |
| 1 | -4 | 4 | 0 | 2 | 2 | 0 |
| 1 | 2 | 5 | -16 | -1 + 2i | -1 - 2i | 4i signed |
Frequently Asked Questions
What is the difference between quadratic roots?
It is the result of subtracting one root from the other.
What is the absolute root difference?
It is the nonnegative distance between two real roots.
What happens when the discriminant equals zero?
Both roots are equal, so their difference equals zero.
Can the calculator handle complex roots?
Yes. Negative discriminants produce complex conjugate roots and an imaginary difference.
Why can the signed difference be negative?
Its sign depends on coefficient a and the selected root ordering.
Can I export the calculation?
Yes. Use the CSV and PDF buttons after calculating.