Rational equation solver
Clear the denominators, solve, then throw out anything that makes a denominator zero.
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What counts as a rational equation?
A rational equation has the variable in a denominator. The standard approach is to multiply everything by the least common denominator (LCD), which turns it into a polynomial equation.
Multiplying by an expression that could be zero can introduce extraneous solutions, answers that satisfy the new equation but not the original. Always check them against the restrictions.
How to solve a rational equation
- Factor each denominator and list the restrictions: values that make any denominator zero.
- Multiply every term on both sides by the LCD.
- Solve the resulting linear or quadratic equation.
- Reject any solution that is one of the restrictions.
- Check the remaining solutions in the original equation.
Frequently asked questions
What is an extraneous solution?
A value that comes out of the algebra but doesn’t satisfy the original equation, usually because it makes a denominator zero. In x/(x − 3) = 3/(x − 3) + 2, the algebra gives x = 3, but x = 3 is not allowed, so there is no solution.
Can I cross-multiply?
Yes, when each side is a single fraction: a/b = c/d becomes ad = bc. It’s a shortcut for multiplying both sides by bd, and the same restrictions apply.
How do I find the LCD?
Factor every denominator and take each different factor the greatest number of times it appears in any one denominator.