What is the LCD method?

The least common denominator is the smallest polynomial expression that every denominator divides into evenly. Multiplying both sides of the equation by the LCD cancels all denominators, leaving a polynomial equation. The solver computes the LCD by taking the product of unique denominator factors.

What are restricted values?

A restricted value is any number that makes a denominator zero. Division by zero is undefined, so these values are excluded from the domain. The solver identifies all restrictions before solving and checks every candidate solution against them. The variable can never equal a restricted value.

What is an extraneous solution?

An extraneous solution is a value that satisfies the polynomial obtained after clearing fractions but violates the original equation by making a denominator zero. Extraneous solutions arise because multiplying by an expression that could be zero is not a reversible operation. Every candidate must be checked against the restricted values.

Why do I need to check solutions in rational equations?

Multiplying both sides by the LCD can introduce roots that did not exist in the original equation. These extraneous roots are valid solutions of the polynomial but invalid for the original rational equation. Checking each candidate against the restricted-value list catches them before they are reported as final answers.

Rational Equation Solver

¹/ₓ

Solve equations with variables in denominators, step by step

Enter equation...

Var

Num

Select an equation type or enter your own rational equation, then click Solve to see the step-by-step solution.

Getting Started

Building Equations with Buttons and Keyboard

Try an Example — Eight Form Templates

Reading the Step-by-Step Solution

The LCD Method — Why It Works

What is a Rational Equation?

The Solving Process Explained

Related Concepts

Getting Started

The solver shows an equation display at the top with a blinking yellow caret marking the cursor position. Click anywhere in the display to place the cursor, or use arrow keys, Home, and End for keyboard navigation. Type directly or use the button panel below.

Three quick experiments:

• Type

1/x + 1/2 = 3/4

and press Enter — the right panel identifies the LCD as

4x

, multiplies through, solves the resulting linear equation, and confirms

x = 4

does not violate the restriction

x \neq 0

.
• Click an example template like "Has Extraneous Root" — the solver finds a candidate root that turns out to make a denominator zero and rejects it.
• Enter

x / (x - 3) = 3 / (x - 3) + 2

to see the extraneous-solution mechanic in action.

The Solve button is disabled until you enter something. Pressing Enter on the display is equivalent to clicking Solve.

Building Equations with Buttons and Keyboard

The button panel groups inputs by type. All inputs can be made either by clicking or typing on the keyboard.

• Variable row —

x

y

n

. The solver picks up whichever variable appears in your equation.
• Number row — digits 0 through 9 and the decimal point.
• Operator row — multiplication, division, plus, minus, equals.
• Power and bracket row —

x^2

shortcut, caret for arbitrary exponents, and parentheses for grouping denominators.

The division symbol always introduces a fraction in the display, with the numerator above and the denominator below. For multi-term denominators wrap them in parentheses:

1/(x + 3)

not

1/x + 3

.

Keyboard shortcuts: type letters and digits directly, use star or slash for multiplication and division (auto-converted on display), and press Enter to solve. Ctrl+Z undoes up to fifty edits back; Backspace deletes the character before the cursor; Delete removes the one after it. The Clear button empties the display entirely; the curved-arrow button steps through the undo stack one edit at a time.

Try an Example — Eight Form Templates

Click the "Try an Example" header to expand the template panel. Each card generates a random equation of that form. Clicking again produces a new random version.

• Simple Reciprocal —

a / x = b

. One variable denominator; cross-multiplication gives

x = a / b

.
• Linear Denominator —

a / (bx + c) = d

. Slightly more complex denominator; same cross-multiplication approach.
• Proportion —

a / (x + b) = c / (x + d)

. Two fractions equated; cross-multiply directly.
• Sum of Fractions —

1/x + 1/(x + a) = b

. Two different denominators; the LCD is their product.
• Has Extraneous Root — constructed so a candidate solution equals a restricted value. Tests the extraneous-check step.
• Quadratic After Clearing —

x + a/x = b

. Multiplying through by

x

yields a quadratic.
• Variable Numerator —

(ax + b) / (cx + d) = e

. Standard rational form with linear numerator and denominator.
• No Solution —

1/(x - a) = 1/(x - a) + b

for

b \neq 0

. Reduces to

0 = b

, a contradiction.

Roughly 80 percent of generated equations produce clean integer or rational solutions; the rest exercise irrational and extraneous cases.

Reading the Step-by-Step Solution

The solution panel shows each algebraic move as a labeled step. The step sequence is fixed for rational equations.

• Identify Restrictions — lists the values of the variable that would make any denominator zero, computed by solving each denominator for zero.
• Find LCD — displays the least common denominator as a polynomial expression.
• Multiply Both Sides by LCD — shows the polynomial equation that results after clearing all fractions.
• Solve Polynomial — routes the cleared equation through the polynomial solver (linear, quadratic, or higher via rational root theorem).
• Candidate Solutions — lists the raw roots of the polynomial before checking.
• Check for Extraneous Solutions — compares each candidate against the restricted values; flags any matches as extraneous.
• Solution — reports the valid roots that remain after the extraneous check.

If every candidate turns out to be extraneous, the equation has no solution and the final answer card displays the contradiction with a cross mark.

The LCD Method — Why It Works

Multiplying both sides of an equation by a nonzero quantity preserves the solution set. The LCD is the smallest expression that turns every denominator into a polynomial, so multiplying by it clears all fractions in one step.

Construction. If the unique denominators in the equation are

d_1, d_2, \ldots, d_k

, the solver builds the LCD as their product

d_1 \cdot d_2 \cdots d_k

(factoring out common factors when feasible). For each fraction

n_i / d_i

, multiplying by the LCD leaves

n_i \cdot (\text{LCD} / d_i)

— a polynomial.

Example. For

\frac{1}{x} + \frac{1}{x + 2} = \frac{3}{4}

, the unique denominators are

x

x + 2

, and

4

, so

\text{LCD} = 4x(x + 2)

. Multiplying every term by

4x(x + 2)

4(x + 2) + 4x = 3x(x + 2)

This is a polynomial equation, solvable by standard methods.

The catch. Multiplying by an expression that can equal zero is not an equivalence operation. If the LCD evaluates to zero at some candidate solution, that candidate was created by the multiplication and is not a true solution of the original equation. This is why the extraneous-check step matters.

For deeper coverage see least common denominator, clearing fractions, and equivalent equations.

What is a Rational Equation?

A rational expression is a ratio of two polynomials:

\frac{P(x)}{Q(x)}

where

P

and

Q

are polynomials and

Q(x) \neq 0

. A rational equation sets one rational expression equal to another, or to a polynomial:

\frac{P_1(x)}{Q_1(x)} + \frac{P_2(x)}{Q_2(x)} + \cdots = R(x)

The defining feature: the variable appears in at least one denominator. This makes the equation fundamentally different from a polynomial equation because the domain is restricted — any value that makes a denominator zero must be excluded.

The graph of

y = P(x) / Q(x)

typically has vertical asymptotes at the restricted values, and may have horizontal or slant asymptotes far from the origin. Rational equations correspond to finding where two such graphs intersect, or where one crosses a constant level.

Rational equations appear in physics (lens formulas, parallel resistors), chemistry (concentrations, reaction rates), economics (average cost per unit), and any setting where two quantities are combined as inverses or ratios.

For deeper coverage see rational expressions, rational functions, and vertical asymptotes.

The Solving Process Explained

Four steps solve any rational equation. The first three are mechanical; the fourth is essential and easily forgotten.

• Step 1: Identify the restricted values. For each denominator, set it equal to zero and solve. These values are excluded from the domain. Skipping this step means missing the check that distinguishes a valid solution from an extraneous one.

• Step 2: Compute the LCD. The least common denominator is the smallest polynomial divisible by every denominator. For distinct denominators with no common factors, the LCD is simply their product.

• Step 3: Multiply both sides by the LCD. Every fraction becomes a polynomial. The resulting equation is a polynomial equation of degree equal to (or less than) the degree of the LCD.

• Step 4: Solve and check. Solve the polynomial equation by standard methods. For each candidate root, verify that it does not equal any restricted value. Reject any root that does — it is extraneous. The remaining roots are the true solutions.

If the polynomial has no real roots, the equation has no solution. If every root is extraneous, the equation also has no solution.

For comprehensive treatment see solving rational equations, extraneous solutions, and cross multiplication.

Related Concepts

Rational Functions — functions of the form

f(x) = P(x) / Q(x)

where

P

and

Q

are polynomials. Have vertical asymptotes at zeros of

Q

that are not also zeros of

P

.

Rational Inequalities — same form as rational equations but with

<

>

\leq

, or

\geq

. Solved by finding zeros of both numerator and denominator, then sign-chart analysis on the intervals.

Vertical Asymptotes — vertical lines

x = a

where the function value tends to

\pm \infty

. Occur at restricted values that are not removable.

Cross Multiplication — for the special form

a/b = c/d

, multiplying gives

ad = bc

. Works only for proportions (two fractions equated, nothing else).

Polynomial Equations — the equation type that results from clearing fractions. The solver routes the cleared equation through the polynomial solver for the final step.

Partial Fractions — a decomposition that runs in the opposite direction: breaking a single rational expression into a sum of simpler fractions. Used in calculus for integration.

Domain Restrictions — the set of values a variable cannot take. For rational expressions, the domain excludes all zeros of the denominator.

Extraneous Roots — candidate solutions introduced by an operation that is not equivalence-preserving. Common in rational and radical equations.