Build the equation from coefficients, then solve fast. Handle quadratics, exclusions, and extraneous roots safely. Clear results, exports, and guidance for every problem set.
| a1 | b1 | c1 | d1 | a2 | b2 | c2 | d2 | k |
|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 1 | -3 | 2 | -1 | 1 | 1 | 0 |
The solver clears denominators by multiplying both sides by (c1x+d1)(c2x+d2). That transforms the rational equation into a polynomial:
Expanding yields Ax² + Bx + C = 0, where:
Candidate roots are then tested in the original rational equation to remove extraneous solutions and any excluded values.
This tool solves equations of the form (a1x+b1)/(c1x+d1) = (a2x+b2)/(c2x+d2) + k, where each numerator and denominator is linear. Clearing denominators produces at most a quadratic, so you get up to two candidate roots. Because denominators may be zero, the solver also reports excluded x values before validating solutions.
For each denominator c1x+d1 and c2x+d2, the forbidden point is x = −d/c when c ≠ 0. These points split the real line into intervals and can create vertical asymptotes. The calculator lists the exclusions and automatically rejects any candidate root that lands within the tolerance of a forbidden point.
Multiplying both sides by (c1x+d1)(c2x+d2) yields Ax²+Bx+C=0. The coefficients depend on all eight inputs plus k, so small changes can shift roots noticeably. The results panel shows A, B, and C so you can reuse them in manual work, check algebra, or compare multiple scenarios.
When you clear denominators, you may introduce extraneous solutions. This calculator substitutes each candidate root back into the original rational equation and reports LHS, RHS, and |LHS−RHS|. A root is marked valid when the absolute error is within the chosen tolerance scaled by 1+|RHS|, making the check robust across magnitudes.
The Plotly chart overlays LHS and RHS across an x range while breaking the lines near exclusions to avoid misleading spikes. Intersections of the two curves correspond to verified solutions, and mismatches highlight extraneous candidates. If k increases, the RHS curve shifts upward by exactly k, which you can see immediately.
Use the CSV export to capture coefficients, exclusions, polynomial terms, and validated roots in a spreadsheet. The PDF export produces a compact one-page record suitable for assignments or audit trails. Keeping the tolerance and rounding digits alongside results helps you reproduce the same validation outcomes later. Consistent notation and verification reduce common mistakes substantially in practice during homework, tutoring, and quick classroom checks for many learners. Store multiple runs, compare coefficient sets, and use the chart to explain clearly why certain intervals cannot contain solutions.
It solves linear-over-linear rational equations on both sides, with an optional constant k added to the right side. After clearing denominators, it handles linear or quadratic cases and validates roots in the original equation.
Clearing denominators can create candidates that satisfy the multiplied equation but not the original fractions. The calculator re-substitutes each candidate into LHS and RHS and rejects any root with error above your tolerance.
Use 1e-8 for typical homework and decimal inputs. If coefficients are very large or very small, slightly increase tolerance to avoid false rejections. For exact integer work, reduce it to 1e-10 for stricter checks.
It plots LHS and RHS across an x range and breaks lines near excluded values to avoid spikes. Intersections indicate solutions. If curves stay separated, there is no valid real solution in the displayed range.
Yes. All coefficient fields accept decimals and negatives. Just remember that denominators can introduce excluded x values, and the solver will automatically remove any root that makes a denominator effectively zero.
Exports include the equation form, coefficient sets, excluded values, polynomial A, B, C, validated real roots, rejected candidates, and the tolerance used. This makes it easier to document work, share results, and reproduce the same verification later.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.