Solving Rational Equations

Rational Equation

Numerator

Denominator

Value (k)

Enter polynomials with integer coefficients: x+1, x^2-4, 2x-3. k can be an integer or fraction like 1/2.

Examples

Numerator 1

Denominator 1

Numerator 2

Denominator 2

Enter polynomials with integer coefficients. Cross-multiplication: num1 × denom2 = num2 × denom1.

Examples

Solution

Enter a rational equation above and press Solve to see all solutions with extraneous solution checking.

Candidate Solutions (before checking)

Extraneous Solution Check (denominator = 0?)

LCD Method: Clearing Denominators

Multiply every term on both sides by the LCD

Step 1: Identify the LCD — the least common denominator of all fractions in the equation.

Step 2: Multiply every term on both sides by the LCD. Each fraction's denominator cancels, leaving a polynomial equation.

Step 3: Expand and collect terms, then solve the resulting linear or quadratic equation.

Step 4: ALWAYS check each solution by substituting back into the original denominators. If any denominator equals zero, that solution is extraneous and must be rejected.

Multiplying both sides by a variable expression can introduce solutions that don't satisfy the original equation — that's why checking is non-negotiable.

What Is an Extraneous Solution?

If denom(x) = 0, then x is EXTRANEOUS

A solution is extraneous if it makes any original denominator equal to zero. It appears to solve the cleared equation but is invalid for the original rational equation.

For example, if you solve and get x = 2, but the original equation has denominator (x − 2), then x = 2 makes the denominator zero — it is extraneous and must be excluded.

Extraneous solutions arise from multiplying by variable expressions.
Always substitute candidates back into original denominators.
If ALL solutions are extraneous, the answer is "No Solution."
Domain restriction: x cannot make any original denominator zero.

Rational equations getting complicated?