Rational Equation Calculator
Solve equations with rational expressions and check for extraneous solutions
Left Side of Equation
Right Side of Equation
How to Solve Rational Equations
What is a Rational Equation?
A rational equation is an equation that contains one or more rational expressions (fractions with polynomials). The key to solving them is eliminating the fractions.
Solution Process
- Find the LCD of all denominators in the equation
- Multiply EVERY term on both sides by the LCD
- Simplify (denominators will cancel out)
- Solve the resulting polynomial equation
- CHECK each solution in the original equation!
- Reject any extraneous solutions that make denominators zero
What are Extraneous Solutions?
Extraneous solutions are answers that emerge during the solving process but don't actually work in the original equation. They occur because multiplying by the LCD can introduce new solutions.
Example: If x = 2 makes a denominator zero in the original equation, then x = 2 is extraneous even if it solves the cleared equation.
Example
Solve: 1/(x - 2) = 3/(x + 1)
- LCD = (x - 2)(x + 1)
- Multiply: 1(x + 1) = 3(x - 2)
- Expand: x + 1 = 3x - 6
- Solve: -2x = -7, so x = 7/2
- Check: x = 7/2 doesn't make any denominator zero ✓
- Solution: x = 7/2
Frequently Asked Questions
Why do we multiply by the LCD?
Multiplying by the LCD eliminates all fractions, converting the rational equation into a simpler polynomial equation that's easier to solve.
Do I always need to check my solutions?
YES! Always check. This is the most critical step. If you skip it, you might include extraneous solutions that don't actually work.
How do I identify extraneous solutions?
A solution is extraneous if it makes any denominator in the original equation equal to zero. Check each solution against all restrictions.
Can a rational equation have no solution?
Yes! If all solutions you find are extraneous, the equation has no solution. Also, some equations algebraically have no solution.
What if there are more than two fractions?
The process is the same! Find the LCD of all denominators, multiply every term by it, then solve.
Can I cross-multiply?
Only if the equation has exactly one fraction on each side: a/b = c/d becomes ad = bc. Otherwise, use the LCD method.
What if the LCD is complicated?
Factor each denominator first. The LCD is the product of all unique factors to their highest powers. It might look messy, but it works!
Can there be multiple valid solutions?
Yes! Quadratic and higher-degree equations can have multiple solutions. Check each one individually for restrictions.