1. Find the domain
Set each original denominator unequal to zero. These restrictions apply throughout the solution.
—
Enter an equation to find its domain.
Rational Equation Solver with Domain Restrictions
See the original domain, cleared equation, real candidate roots, rejected values, final solution set, and verification.
Enter the rational equation
Solution
Real solution set
Cleared equation—
Candidate roots—
Rejected values—
Verification—
Advertisement
Step-by-step solution
The domain and algebra steps will appear here.
Advertisement
How to solve rational equations safely
2. Clear denominators
For P/Q = R/S, multiply by QS on the allowed domain to obtain PS = RQ.
3. Solve candidates
Move all terms to one side, factor when possible, and solve the resulting polynomial equation.
4. Reject and verify
Discard every root that zeros an original denominator, then check the remaining values in the original equation.
Core rule
P(x)/Q(x) = R(x)/S(x) ⟹ P(x)S(x) = R(x)Q(x), provided Q(x)S(x) ≠ 0
Clearing denominators is equivalence-preserving only after the original domain has been stated.
Examples and special cases
Excluded candidate
(x+1)/(x−2)=3/(x−2) clears to x+1=3, suggesting x=2. But x=2 is outside the original domain, so there is no solution.
One valid solution
1/x=2/(x+3) has restrictions x≠0,−3. Cross multiplication gives x+3=2x, so x=3.
Identity on a restricted domain
(x²−1)/(x−1)=x+1 is true for every real x except 1. Cancellation does not restore the missing input.
No real roots
If clearing denominators produces x²+1=0, the rational equation has no real solution, though it has complex polynomial roots.
Calculation method and limits
Parsing, cross multiplication, and rational-root checks use reduced BigInt fractions. Linear roots, rational roots, and roots of quadratic factors are shown exactly. Any unresolved real roots of degree 3–8 are computed numerically, labeled as approximations, and checked against the original equation and denominators.
The solver works over the real numbers and accepts polynomial numerators and denominators in one variable through degree 8. It does not solve nested rational expressions, radicals, functions, inequalities, or systems. Numerical methods can lose reliability for extremely ill-conditioned polynomials or nearly repeated roots. Last reviewed: 4 August 2026.
Rational equation FAQs
How do you solve a rational equation?
Find the original denominator restrictions, multiply through by a common denominator, solve the resulting polynomial, and reject any excluded candidate.
Why find restrictions before simplifying?
Because a canceled denominator factor still marks a value where the original equation was undefined. Simplification cannot add that value back to the domain.
Can cross multiplication create an extraneous answer?
It can produce a candidate outside the original domain. Cross multiplication is equivalent to the original equation only where both original denominators are nonzero.
Can the answer be all real numbers?
Yes, but only the real numbers in the original domain. An identity with a denominator restriction is not true at excluded values.
Does the solver show complex solutions?
No. This page reports real solutions. Use the polynomial root calculator for real and complex zeros of a polynomial.
Are my equation entries private?
Yes. The equation is processed locally by JavaScript in your browser and is not sent or stored by this tool.