Calculator Inputs
Use the form below to model equations in the form (a1x+b1)/(c1x+d1) + (a2x+b2)/(c2x+d2) = (a3x+b3)/(c3x+d3) + k. Set both numerator coefficients to zero to disable an optional term.
Example Data Table
This sample matches the default inputs on first load.
| Item | Value | Meaning |
|---|---|---|
| Equation | 1/(x-1) + 1/(x+1) = 1 |
Two rational terms on the left and a constant on the right. |
| Restrictions | x ≠ 1, -1 |
These values make a denominator equal to zero. |
| Cleared Form | x^2 - 2x - 1 = 0 |
Multiply both sides by (x-1)(x+1). |
| Valid Solutions | x ≈ -0.414214, 2.414214 |
Both roots satisfy the original equation and avoid restrictions. |
Formula Used
For active terms, the calculator uses the general form:
(a1x+b1)/(c1x+d1) + (a2x+b2)/(c2x+d2) = (a3x+b3)/(c3x+d3) + k
Each denominator creates a restriction:
cix + di ≠ 0
The calculator multiplies both sides by the product of all active denominators, expands the expression, and converts the rational equation into a polynomial equation.
The working cleared form is:
(a1x+b1)D2D3 + (a2x+b2)D1D3 - (a3x+b3)D1D2 - kD1D2D3 = 0
Here, each Di stands for an active denominator such as (cix+di). Real roots are then tested in the original equation to remove extraneous solutions.
How to Use This Calculator
- Enter the numerator and denominator coefficients for each active rational term.
- Set both numerator coefficients of an optional term to zero when you do not need that term.
- Enter the constant
kfor the right side. - Choose a graph range and the number of plotted points.
- Click Solve Equation to see the verified result above the form.
- Review restrictions, the working LCD, candidate roots, rejected roots, and the final checked solution set.
- Use the CSV or PDF buttons to export the calculated summary.
FAQs
1. What is a rational equation?
A rational equation contains one or more fractions with algebraic expressions in the denominator. Solving it usually requires clearing denominators and checking for invalid roots.
2. Why does the calculator show restrictions first?
Restrictions matter because denominator values cannot become zero. Any candidate root that violates a restriction must be rejected, even if it solves the cleared polynomial.
3. What is the LCD in this solver?
The LCD is the working product of all active linear denominators. Multiplying by it removes fractions and turns the rational equation into a polynomial equation.
4. Why can a candidate root be rejected?
Some roots appear after clearing denominators but fail the original equation. These are extraneous roots, and the calculator removes them after direct substitution.
5. Can this tool show no-solution or identity cases?
Yes. It reports no valid real solution when every candidate fails. It also reports infinitely many real solutions when the cleared equation becomes an identity.
6. What happens if I set a term numerator to zero?
That optional rational term is treated as disabled. The calculator then ignores its denominator when building restrictions, the graph, and the solved polynomial.
7. How should I choose the graph range?
Pick a range that includes expected intersections and nearby asymptotes. A wider range is useful for exploration, while a tighter range gives clearer local detail.
8. Are the answers exact or decimal?
This calculator reports verified decimal roots. The displayed values are rounded to your chosen precision after the original equation is checked numerically.