What is a rational equation?

A rational equation is an equation that contains at least one fraction whose numerator or denominator is a polynomial. For example, 3/(x+2) = 5 or 2/(x-1) + 1/(x+3) = 4 are rational equations. The key challenge is that the variable appears in the denominator, requiring careful handling of excluded values.

How do you find the LCD of rational expressions?

To find the Least Common Denominator (LCD) of rational expressions: (1) Factor each denominator completely. (2) List all unique factors. (3) For repeated factors, use the highest power that appears. (4) Multiply all these factors together. For example, the LCD of 1/(x+2) and 1/(x-3) is (x+2)(x-3).

What are excluded values in rational equations?

Excluded values (also called restrictions) are values of the variable that make any denominator equal to zero, making the expression undefined. For example, in 5/(x-3), the excluded value is x=3 because division by zero is undefined. Any solution to a rational equation must be checked against these excluded values.

What is an extraneous solution?

An extraneous solution is a value that satisfies the transformed equation (after clearing denominators) but does NOT satisfy the original rational equation — typically because it is an excluded value that makes a denominator equal to zero. Always substitute your answer back into the original equation to verify it is valid and not extraneous.

How do work rate problems use rational equations?

Work rate problems use the formula 1/a + 1/b = 1/t, where a is the time for person A to complete the job alone, b is the time for person B alone, and t is the combined time. This is a rational equation because the unknowns appear in denominators. Solving it involves finding the LCD (usually abt) and multiplying through to clear all fractions.

What is cross multiplication and when can you use it?

Cross multiplication is a shortcut for equations of the form a/b = c/d: multiply the numerator of each side by the denominator of the other to get ad = bc. This works ONLY when there is exactly one fraction on each side. For equations with sums or differences of fractions, you must use the full LCD method instead.

Can a rational equation have no solution?

Yes. A rational equation has no solution when every candidate solution is extraneous (an excluded value). This is different from a quadratic with no real roots — here the algebraic solution exists but is rejected due to the domain restriction.

What happens when clearing denominators produces a quadratic?

When you multiply both sides of a rational equation by the LCD, the resulting polynomial equation may be quadratic (degree 2). You solve it using factoring or the quadratic formula to find two candidate solutions. Then check each against the excluded values — some or all may be extraneous.

Rational Equation Solver — Solve Equations with Fractions Step by Step

Select Equation Type

Form: a / (b·x + c) = d

a (numerator)

b (coeff of x)

c (constant)

d (RHS value)

How to Use This Solver

1
Choose equation type — pick the tab that matches your equation's structure: single fraction, fraction equals fraction, sum, quadratic result, or work rate.
2
Enter coefficients — fill each field according to the labeled form. Decimals and negative values are accepted. Click "Try Example" to see a pre-filled sample.
3
Click Solve — the solver identifies the LCD, clears all fractions, expands, and solves step by step.
4
Review excluded values — values that make any denominator zero are flagged before solving begins.
5
Check verification step — each candidate answer is substituted back into the original equation and checked for extraneous status automatically.

What Are Rational Equations?

A rational equation is any equation that contains at least one rational expression — a fraction whose numerator or denominator (or both) contains a variable. Examples include 3/(x+2) = 5, 1/x + 2/(x-1) = 3, and the classic work-rate formula 1/a + 1/b = 1/t. The defining feature is that the unknown variable appears in at least one denominator, making the expression undefined at certain values.

Rational equations arise naturally across science and everyday life: calculating the combined speed of two workers sharing a task, finding the time for parallel pipes to fill a tank, determining mixture concentrations, and modeling electrical circuits with parallel resistors (1/R_total = 1/R1 + 1/R2). Mastering their solution connects arithmetic fraction skills to full polynomial algebra and is a gateway to advanced algebraic reasoning.

This solver handles five common structural forms, walks through every algebraic transformation, flags excluded values before any computation, and automatically detects extraneous solutions — going well beyond simple cross-multiplication tools.

The LCD Method — The Standard Approach

The most systematic way to solve rational equations is the Least Common Denominator (LCD) method. The strategy: identify the LCD of every fraction in the equation, then multiply every term on both sides by that LCD. This "clears" all fractions in one step, leaving a standard polynomial equation (linear or quadratic) which is straightforward to solve. The LCD must include every distinct factor from every denominator, each raised to its highest occurring power.

Excluded Values and Extraneous Solutions

A critical step unique to rational equations is identifying excluded values — numbers that make any denominator equal zero. These must be excluded from the solution set because division by zero is undefined. After solving, each candidate solution must be tested: if it equals an excluded value, it is an extraneous solution and must be discarded. Forgetting this final check is one of the most common algebra errors at every level.

Formulas Reference

Cross Multiplication

a/b = c/d → ad = bc

Works only when one fraction equals one fraction (no sums).

LCD Method

Multiply all terms by LCD

Use when adding/subtracting fractions or more than 2 fraction terms.

Work Rate Formula

1/a + 1/b = 1/t

Combined rate = sum of individual rates. Solved: t = ab/(a+b).

Quadratic Formula

x = (-b ± √(b²‑4ac)) / 2a

Applied when clearing fractions yields a degree-2 polynomial.

Worked Example 1 — Single Fraction Equals a Number

Problem: Solve 6 / (2x − 4) = 3

Excluded values: Set denominator = 0: 2x − 4 = 0 → x = 2. Domain restriction: x ≠ 2.
LCD = (2x − 4). Multiply both sides: 6 = 3(2x − 4)
Expand: 6 = 6x − 12
Solve: 6x = 18 → x = 3
Check: x = 3 ≠ 2, so it is not excluded. Substitute: 6/(2·3−4) = 6/2 = 3 ✓ Valid solution.

Worked Example 2 — Sum of Fractions Equals a Number

Problem: Solve 2/(x+3) + 3/(x−2) = 2

Excluded values: x ≠ −3 and x ≠ 2
LCD = (x+3)(x−2). Multiply each term by LCD.
Expand LHS: 2(x−2) + 3(x+3) = 2x−4 + 3x+9 = 5x+5
Expand RHS: 2(x+3)(x−2) = 2(x²+x−6) = 2x²+2x−12
Rearrange: 2x²+2x−12 − 5x−5 = 0 → 2x²−3x−17 = 0
Quadratic formula gives two candidates; check both against excluded values x ≠ −3, x ≠ 2.

Rational Equation Solver