Calculator
Example Data Table
| Equation | Discriminant | Root Type | Roots |
|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | Two distinct real roots | 3, 2 |
| x² + 2x + 5 = 0 | -16 | Two imaginary roots | -1 + 2i, -1 - 2i |
| 2x² + 4x + 2 = 0 | 0 | One repeated real root | -1, -1 |
| 3x² - x - 2 = 0 | 25 | Two distinct real roots | 1, -0.666667 |
Formula Used
For any quadratic equation in the form ax² + bx + c = 0, first calculate the discriminant:
D = b² - 4ac
Then apply the quadratic formula:
x = (-b ± √D) / (2a)
If D > 0, the roots are real and different.
If D = 0, the roots are real and equal.
If D < 0, the roots are imaginary and appear as complex conjugates:
x = -b / (2a) ± (√|D| / 2a)i
How to Use This Calculator
Enter the coefficient values for a, b, and c.
Choose the decimal precision you want in the final output.
Press Calculate Roots to show the result section above the form.
Review the discriminant, the root type, both roots, and the step breakdown.
Use the CSV button to export the current result as a spreadsheet-friendly file.
Use the PDF button to save the visible report as a downloadable document.
About Real and Imaginary Roots
Why this topic matters
A quadratic equation appears in algebra, geometry, finance, and physics. The roots show where the expression becomes zero. That makes root analysis very useful. You can test trajectories, optimize models, and study graphs with one equation. A real and imaginary roots calculator saves time and reduces small mistakes. It also makes the classification process clear.
How the discriminant guides the answer
The key value is the discriminant. It is found from b² - 4ac. This single number tells you the nature of the roots. A positive discriminant gives two real roots. A zero discriminant gives one repeated real root. A negative discriminant gives two imaginary roots. Those imaginary roots are also called complex conjugates. They always come as a pair.
What real roots and imaginary roots mean
Real roots can be placed on the number line. They can be integers, fractions, or decimals. Imaginary roots include the symbol i, where i² equals -1. These roots do not sit on the real number line. Still, they are valid mathematical solutions. They are important in signal analysis, control systems, advanced algebra, and polynomial theory.
Why a calculator helps
Manual work is fine for simple equations. It becomes slower when coefficients are negative, decimal based, or large. This calculator automates the hard part. It finds the discriminant, classifies the root type, and displays both solutions. It also shows the sum of roots, product of roots, and axis of symmetry. These values help when checking graph behavior or factor relationships.
Practical value for students and professionals
Students can use the tool for homework practice and exam review. Teachers can use it for classroom examples. Engineers and analysts can use it during fast verification. The export options help with records and reporting. The example table also makes pattern recognition easier. When you compare multiple equations, you quickly see how coefficient changes affect root behavior and equation structure.
Frequently Asked Questions
1. What does this calculator solve?
It solves quadratic equations of the form ax² + bx + c = 0. It identifies whether the roots are real, repeated, or imaginary. It also shows both solutions.
2. Why must a not equal zero?
If a equals zero, the equation is no longer quadratic. It becomes linear. The quadratic formula and discriminant rules used here would no longer apply.
3. What are imaginary roots?
Imaginary roots happen when the discriminant is negative. The square root of a negative number introduces i. The final answers appear as complex conjugates.
4. What happens when the discriminant is zero?
A zero discriminant means both roots are the same. This is called a repeated root. The parabola touches the x-axis at one point only.
5. Can I use decimal coefficients?
Yes. The calculator accepts integers and decimals for a, b, and c. It then computes the roots using your selected decimal precision.
6. What is the exact form line for?
It shows the root expression in formula style. This helps you see the structure of the solution, especially before rounding the decimal output.
7. Why are sum and product of roots included?
These values help verify the answer. For a quadratic, the sum is -b/a and the product is c/a. They are useful for quick checking.
8. Can I save my result?
Yes. After calculation, you can export the current report as CSV or PDF. This helps with assignments, notes, and calculation records.