Calculate the solution count
Enter coefficients for a standard equation: ax2 + bx + c = 0.
Formula used
For a quadratic equation ax2 + bx + c = 0, the calculator evaluates the discriminant.
When D is positive, there are two distinct real solutions. When D is zero, one repeated real solution exists. When D is negative, there are no real solutions but two complex solutions. Linear and constant cases are handled before this test.
How to use this calculator
- Rewrite the equation so one side equals zero.
- Enter the coefficients a, b, and c.
- Use zero for any missing term.
- Select real or complex numbers as the solution domain.
- Choose a display precision and press the calculation button.
- Read the classification, solution count, discriminant, and listed roots.
Example data table
| Equation | a | b | c | Discriminant | Real solutions |
|---|---|---|---|---|---|
| x2 − 5x + 6 = 0 | 1 | −5 | 6 | 1 | 2 |
| x2 − 4x + 4 = 0 | 1 | −4 | 4 | 0 | 1 repeated |
| x2 + 2x + 5 = 0 | 1 | 2 | 5 | −16 | 0 |
| 3x − 9 = 0 | 0 | 3 | −9 | Not used | 1 |
Understanding Solution Counts
An equation solution makes the statement true. The count depends on the equation type. It also depends on the chosen number system. A linear equation usually has one solution. A quadratic equation can have several outcomes. This calculator examines the coefficients before selecting the result. It explains the result clearly.
Linear and Constant Cases
Start with the standard expression ax² + bx + c = 0. When a equals zero, the expression is no longer quadratic. When b remains nonzero, one linear solution exists. When both a and b equal zero, inspect c. A nonzero c gives no solution. A zero c makes every allowed number a solution.
The Discriminant Test
When a is nonzero, use the discriminant. The discriminant is b² − 4ac. Its sign identifies the real solution count. A positive value produces two different real roots. A zero value produces one repeated real root. A negative value produces no real roots. These rules make quadratic classification dependable.
Real and Complex Domains
The selected domain changes the displayed answer. In the real domain, negative discriminants have zero solutions. In the complex domain, those equations have two complex roots. A quadratic still has two roots when multiplicity is included. A repeated root counts twice by multiplicity. It remains one distinct root.
Why Coefficients Matter
Small coefficient changes can alter the answer. Changing c may move the graph across the horizontal axis. Changing b can shift the vertex and roots. Changing a can reverse the parabola. Enter values carefully. Decimal values are accepted. The calculator treats tiny discriminants as zero only within a numerical tolerance.
Reading the Result
The result panel reports the equation type first. It then shows the discriminant when appropriate. You will see the number of distinct solutions. You will also see roots when they can be calculated. Complex roots include an imaginary component. Infinite-solution cases receive a separate explanation. This helps prevent confusing special cases with errors.
Useful Checks Before Solving
Confirm each coefficient matches the original equation. Move every term to one side first. Set the equation equal to zero. Use zero for missing terms. Select real numbers for typical school algebra questions. Select complex numbers when imaginary roots are allowed. Increase decimal precision when your coefficients contain many digits.
Practical Uses
Solution counts support graphing and algebra checks. They help identify intersections between a curve and an axis. They also reveal whether a model has feasible real answers. Engineers, students, and analysts use these checks. The result does not replace equation setup. It verifies what the completed expression can produce.
Limits and Interpretation
The tool classifies equations in standard polynomial form. It does not solve inequalities or equations with unknown denominators. Transform those expressions first. Extraneous roots can appear after squaring. Check every reported root in the original equation. That final check protects your work from hidden restrictions and rounding effects.
Frequently asked questions
1. What does this calculator determine?
It determines how many solutions an equation has in the selected domain. It also identifies linear, quadratic, constant, repeated-root, and no-solution cases. Quadratic results include the discriminant and roots when available.
2. What equation form should I enter?
Enter an equation in the form ax² + bx + c = 0. Move all terms to one side first. Then enter the coefficient beside each term. Use zero when a term is missing.
3. What happens when a equals zero?
The calculator changes from a quadratic check to a linear or constant check. If b is not zero, one linear solution exists. If both a and b are zero, c decides whether no solutions or infinitely many solutions exist.
4. Why is the discriminant important?
The discriminant reveals the real-root pattern of a quadratic equation. Positive means two real roots. Zero means one repeated real root. Negative means no real roots. It is calculated as b² − 4ac.
5. Why can a quadratic have one solution?
A quadratic has one distinct real solution when its discriminant is zero. The graph touches the horizontal axis once. Algebraically, the same root appears twice, so it has multiplicity two.
6. What is the difference between real and complex domains?
The real domain includes ordinary numbers on the number line. The complex domain also allows numbers containing i. A negative discriminant gives zero real solutions but two complex solutions.
7. Does the calculator accept decimal coefficients?
Yes. Enter integers, decimals, or scientific notation accepted by your browser. Choose more decimal places when you want longer root values. Very small calculation differences are treated with numerical tolerance.
8. What does multiplicity mean?
Multiplicity tells how many times a root repeats. A repeated quadratic root has multiplicity two. It counts as one distinct solution but two solutions when repetitions are included.
9. Can this tool solve inequalities?
No. This calculator classifies equations equal to zero. Inequalities need sign analysis, interval testing, or graphing. Rewrite the expression as an equation only when you need its root count.
10. Why should I verify the roots?
Verification catches copied coefficients, rounding differences, and extraneous roots created during earlier algebra. Substitute each reported root into the original equation, especially after squaring or clearing denominators.
11. Can I save the result?
Yes. Download a CSV report for spreadsheet use. You can also use the print button and select Save as PDF in the browser dialog. The copy button provides a quick text summary.