Your combined result, method, checks, and restrictions will appear here.
Formula used
Linear equations isolate the variable using inverse operations. Quadratics use the discriminant and quadratic formula. Systems compare coefficient and augmented matrix ranks.
Polynomial roots use a complex numerical iteration. Trigonometric modes enumerate periodic solutions inside your interval. Intersection mode detects crossings and near-tangent contacts.
How to use
- Select the equation or system type.
- Enter every coefficient, expression, or interval value.
- Choose precision and root-counting preferences.
- Press Calculate solutions.
- Review counts, values, restrictions, checks, and graph.
- Copy, export, or print the combined result.
Use an asterisk for multiplication inside expressions, such as 2*x.
Example data
| Mode | Example | Number of solutions | Classification |
|---|---|---|---|
| Linear | 2x + 3 = 11 | 1 | Unique real solution |
| Quadratic | x² - 5x + 6 = 0 | 2 | Two distinct real roots |
| Quadratic | x² + 4x + 4 = 0 | 1 distinct | Repeated real root |
| System | x + y = 4; 2x + 2y = 8 | Infinite | Dependent system |
| Absolute | |2x - 1| = 5 | 2 | Two real solutions |
| Intersection | x² and 4 | 2 | Two graph intersections |
Calculation history
Frequently asked questions
What does infinitely many solutions mean?
Every valid variable value satisfies the equation. Dependent systems describe the same line or plane. Domain restrictions can still exclude particular values.
How does the discriminant classify quadratic roots?
A positive discriminant gives two real roots. Zero gives one repeated real root. A negative value gives two complex conjugates.
Are repeated roots counted once or several times?
Distinct counting lists each value once. Multiplicity counting repeats roots by their algebraic multiplicity. You can change this behaviour above.
Why can a proposed solution be rejected?
Transformations can create extraneous values. Denominators and logarithms also impose domain limits. Verification checks the original equation again.
How accurate are polynomial roots?
Higher-degree roots use numerical iteration. Displayed accuracy follows your chosen decimal precision. Ill-conditioned polynomials may need independent verification.
Can the calculator find complex roots?
Polynomial and quadratic modes report complex roots. Select real-only output to hide them. Counts remain explained inside the calculation details.
How are graph intersections detected?
The interval is sampled for sign changes. Bisection refines crossings to stable values. Near-tangent contacts receive an additional numerical check.
Which expression syntax is supported?
Use x, numbers, parentheses, and standard operators. Supported functions include sine, cosine, logarithms, and square roots. Write multiplication explicitly using an asterisk.
Does this replace formal mathematical proof?
No numerical calculator replaces rigorous proof. Results support checking, exploration, and routine calculations. Important work should receive independent review.