Enter values for the equation form below: (Ax + B) / (Cx + D) = (Ex + F) / (Gx + H)
| A | B | C | D | E | F | G | H | Example Equation | Restrictions | Valid Roots |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 1 | -1 | 1 | -4 | 1 | 2 | (2x + 3)/(x - 1) = (x - 4)/(x + 2) | x ≠ 1, -2 | -0.169049, -11.830952 |
| 3 | -1 | 1 | 4 | 2 | 5 | 1 | -3 | (3x - 1)/(x + 4) = (2x + 5)/(x - 3) | x ≠ -4, 3 | -4.765565, 13.765565 |
Starting equation: (Ax + B) / (Cx + D) = (Ex + F) / (Gx + H)
Restrictions: Cx + D ≠ 0 and Gx + H ≠ 0
Cross-multiplication: (Ax + B)(Gx + H) = (Ex + F)(Cx + D)
Expanded standard form: ax² + bx + c = 0
Quadratic coefficients:
- a = AG - EC
- b = AH + BG - ED - FC
- c = BH - FD
Discriminant: Δ = b² - 4ac
- If Δ < 0, no real roots appear.
- If Δ = 0, one repeated real root appears.
- If Δ > 0, two real roots appear.
Final validation: Each candidate root is substituted back into the original rational equation.
Any root that makes a denominator zero is rejected.
- Enter the eight coefficients for both rational expressions.
- Use the displayed structure to match each coefficient correctly.
- Click Solve Equation to generate the result panel.
- Review restrictions before accepting any candidate root.
- Check the graph to see where both expressions intersect.
- Use CSV or PDF export when you need a report.
1) What form does this calculator solve?
It solves equations written as (Ax + B)/(Cx + D) = (Ex + F)/(Gx + H). Clearing denominators produces a quadratic, linear, identity, or inconsistent equation.
2) Why are some candidate roots rejected?
A candidate root is rejected when it makes either denominator zero. Rational equations always exclude values that break the original domain.
3) What if the x² term disappears?
Then the cleared equation becomes linear instead of quadratic. The calculator detects that case automatically and solves the simpler equation.
4) Can I use decimals and negative numbers?
Yes. All coefficient fields accept decimals, integers, and negative values. That makes the tool suitable for classroom, homework, and quick checking.
5) What does the discriminant tell me?
The discriminant shows how many real candidate roots the quadratic has. Negative means none, zero means one repeated root, and positive means two real roots.
6) Why does the graph contain gaps?
Gaps appear near vertical asymptotes. Those x-values make a denominator zero, so the rational expression is undefined there.
7) Can this calculator return no real solution?
Yes. That happens when the discriminant is negative or when every real candidate root is removed by the domain restrictions.
8) Can the answer be infinitely many solutions?
Yes. If clearing denominators reduces the equation to 0 = 0, then every allowed real number satisfies the equation except restricted values.