Why must denominator restrictions be found first?

A rational expression is undefined wherever its denominator is zero. Clearing denominators cannot make those excluded inputs valid.

Can a rational equation have an extraneous solution?

Yes. The cleared polynomial may have a root that was excluded from the original equation. That value is a candidate, not a solution.

Can a rational equation have infinitely many solutions?

Yes. If both sides become the same expression, every real number in the original shared domain is a solution.

What input does this solver accept?

Each of the four fields accepts a polynomial in x through degree 8, with parentheses, implicit multiplication, nonnegative integer powers, and integer, decimal, or fractional coefficients.

Are higher-degree answers exact?

Rational roots and roots of linear or quadratic factors are exact. Remaining real roots of higher-degree polynomials are labeled as numerical approximations.

Rational Equation Solver with Domain Restrictions

How to solve rational equations safely

1. Find the domain

Set each original denominator unequal to zero. These restrictions apply throughout the solution.

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.