How do you solve a rational equation?

Find the values that make any denominator zero and set them aside. Multiply every term on both sides by the least common denominator so the fractions disappear, solve the polynomial equation that's left, and then throw out any answer that was on your excluded list.

What is an extraneous solution in a rational equation?

It's a value that solves the equation you got after clearing the fractions but not the original one, because it makes a denominator zero. Multiplying by an expression that can equal zero is what lets these false answers in.

Can a rational equation have no solution?

Yes. If every answer you find is extraneous, or the variable cancels and leaves a false statement such as 0 = 1, there is no solution. x/(x − 3) = 3/(x − 3) + 2 is an example: the only candidate is x = 3, which is excluded.

When can you cross multiply a rational equation?

Only when the equation is a single fraction equal to a single fraction, like 2/(x − 1) = 3/(x + 2). With more terms on either side, multiply everything by the LCD instead.

How do you find the LCD of rational expressions?

Factor every denominator, then take each different factor the greatest number of times it appears in any one denominator. For x − 1, x + 1 and x² − 1 = (x − 1)(x + 1), the LCD is (x − 1)(x + 1).

Does the calculator handle equations that become quadratic?

Yes. If clearing the fractions leaves a quadratic, it factors it when it can and uses the quadratic formula when it can't, giving exact answers such as (1 ± √5)/2. Higher-degree leftovers are solved numerically.

Rational Equations Calculator: Solve With Steps

Rational equations have a reputation for being fiddly, and it’s mostly deserved. The algebra itself is ordinary; the trouble is that there are four or five places to slip, and one of them, the extraneous solution, produces an answer that looks perfectly fine until you plug it back in. This rational equations calculator solves the equation the way you’d be expected to show it in class, with each step written out, and it tells you plainly when an answer has to be thrown away.

If a over (x minus p) equals b over (x minus q), then a times (x minus q) equals b times (x minus p), with x not equal to p or q — Clearing the denominators turns a rational equation into a polynomial one

How to use the rational equations calculator

Type the equation with one equals sign. Put every denominator in brackets: 3/(x + 2), not 3/x + 2, which means .
Use ^ for powers, as in 4/(x^2 - 1). Numbers, fractions and decimals all work.
Read the answer at the top. Excluded values and the LCD are listed underneath, along with any answer that was rejected as extraneous.
Look at the graph. The blue and orange curves are the two sides of the equation. Solutions are where they cross, and dashed lines mark the x-values that are off limits.

The method, step by step

List the restrictions. Any that makes a denominator zero can never be a solution.
Find the LCD of all the denominators, factored.
Multiply every term on both sides by the LCD. Each fraction cancels down to a polynomial.
Solve the equation that’s left. It’s usually linear or quadratic.
Check every answer against the restrictions from step 1.

Step 5 is the one people skip, and it’s the one teachers look for.

Worked example: 2/(x − 1) = 3/(x + 2)

This is the calculator’s first example.

Step 1 — restrictions. The denominators are zero at and , so neither can be an answer.

Step 2 — clear the fractions. With a single fraction on each side, multiplying by the LCD is the same as cross-multiplying:

Step 3 — solve.

Step 4 — check. 7 isn’t excluded, and both sides come out equal: . So .

Worked example with more than two fractions

Solve .

Step 1 — factor the denominators. , so the restrictions are and , and the LCD is .

Step 2 — multiply every term by the LCD. Each term keeps only the factors its own denominator was missing:

Step 3 — solve.

Step 4 — check. isn’t 1 or −1, so it stands. Cross-multiplying wouldn’t have worked here, because the left side has two separate fractions.

Extraneous solutions, and why they appear

When you multiply both sides by the LCD, you’re multiplying by an expression that equals zero at the excluded values. Multiplying by zero makes any equation true, so the new equation can pick up solutions the original never had. They aren’t mistakes in your algebra; they’re a side effect of the method, and the check is how you remove them.

A number line for 1 + 2/(x − 1) = 2/(x² − x) with open circles at the excluded values 0 and 1, a green dot at the solution x = −2, and a red cross at x = 1 marking it as extraneous because it lands on an excluded value; clearing the fractions gives x² + x − 2 = 0, which factors as (x + 2)(x − 1) = 0 — Both roots solve the polynomial, but one of them lands on a hole

Take . The denominators factor as and , so and . Multiplying by the LCD :

The candidates are and . But is excluded, so the only solution is . If you’d stopped at the factoring, you’d have handed in a wrong answer that looks right.

Special cases the calculator reports

What happens	Example	Result
Every candidate is excluded		no solution
The variable cancels, leaving a false statement		no solution
The variable cancels, leaving a true statement	both sides identical	every x except the restrictions
The leftover equation is quadratic

Solving versus simplifying

People often search for a “simplifying rational equations calculator” when they mean one of two different jobs, and it helps to know which one you have.

An expression has no equals sign, like . You simplify it by factoring and cancelling: , with . The answer is another expression.
An equation has an equals sign, like . You solve it, and the answer is a value of . Here, clearing the fraction gives , so and or . The is excluded, so .

This calculator does the second job. For the first, see simplifying rational expressions. The two meet in the middle: simplifying each side first often makes the LCD smaller and the solving shorter.

Where rational equations come from

They turn up whenever a rate is involved. Work problems are the classic case: if one pipe fills a tank in 4 hours and another in 12, together they fill of it per hour, so the time solves . Type 1/4 + 1/12 = 1/x and the calculator gives 3 hours. Speed and distance problems, where time is distance divided by speed, and inverse variation problems lead to the same kind of equation.

Common mistakes

Multiplying only the fractions by the LCD. Every term gets multiplied, including whole numbers like the 1 in .
Cross-multiplying with more than one fraction on a side. It only works for one fraction equal to one fraction.
Skipping the factoring. and share a factor. Missing it gives a bigger LCD and a messier equation.
Not checking the answers. Extraneous solutions look exactly like real ones.
Typing 3/x + 2 for . Brackets around the denominator are what tell the calculator, and your own algebra, where the fraction ends.

The rational equations lesson covers the method in more depth, and the rational equations practice problems make a good solving rational equations worksheet, with answers checked as you go.

Rational Equations Calculator