요금제

Rational Equation Calculator

This rational equation calculator solves equations with fractions that have x in the denominator, such as 2/(x − 1) = 3/(x + 2). Sumzo clears the fractions, solves, and checks every answer against the excluded values.

구할 문자(x, y, r…)

팁: √, x², 분수 같은 기호는 수학 키보드로 입력하세요. 문제의 사진, 스크린샷 또는 PDF도 첨부할 수 있습니다(최대 4개 파일).

예제 풀어 보기:
설명

수학 키보드

키는 커서 위치에 입력됩니다. 먼저 일부를 선택하면 감쌀 수 있습니다 — 예: x + 1을 선택하고 √를 누르세요.

How to solve the rational equation

  1. Type the equation with brackets around each denominator: 2/(x - 1), not 2/x - 1.
  2. Press Solve. Sumzo finds the values x can’t take, multiplies both sides by the common denominator, and solves what is left.
  3. Every solution is substituted back; any that make a denominator zero are rejected as extraneous.

Worked examples

Easy: 6x=3\frac{6}{x} = 3

  1. Excluded value: x≠0x \neq 0.
  2. Multiply both sides by xx: 6=3x6 = 3x, so x=2x = 2.
  3. Check: 62=3\frac{6}{2} = 3 ✓.

Medium: 2x−1=3x+2\frac{2}{x - 1} = \frac{3}{x + 2}

  1. Excluded values: x≠1x \neq 1 and x≠−2x \neq -2.
  2. Cross-multiply: 2(x+2)=3(x−1)2(x + 2) = 3(x - 1), so 2x+4=3x−32x + 4 = 3x - 3 and x=7x = 7.
  3. Check: 26=13\frac{2}{6} = \frac{1}{3} and 39=13\frac{3}{9} = \frac{1}{3} ✓.

Hard: an extraneous solution

  1. Solve x2x−1=1x−1\frac{x^2}{x - 1} = \frac{1}{x - 1}. Excluded value: x≠1x \neq 1.
  2. Multiply both sides by x−1x - 1: x2=1x^2 = 1, so x=1x = 1 or x=−1x = -1.
  3. x=1x = 1 makes the denominator zero, so it is extraneous and rejected.
  4. Check x=−1x = -1: both sides equal 1−2=−12\frac{1}{-2} = -\frac{1}{2} ✓. The only solution is x=−1x = -1.

Common mistakes with rational equations

  • Not checking the answers: a value that makes any denominator zero is never a solution.
  • Multiplying only some terms by the common denominator — every term on both sides must be multiplied.
  • Typing 1/x + 1 when you mean 1/(x + 1): brackets matter.
  • Cross-multiplying when there are more than two fractions; use the common denominator instead.

자주 묻는 질문

What is a rational equation?

An equation with at least one fraction whose denominator contains the variable, such as 2x−1=3x+2\frac{2}{x - 1} = \frac{3}{x + 2}.

How does the calculator solve it?

Write down the excluded values, multiply every term by the lowest common denominator to clear the fractions, solve the resulting equation, then check each answer in the original.

What is an extraneous solution?

A value that solves the equation after the fractions are cleared, but makes a denominator zero in the original. In x2x−1=1x−1\frac{x^2}{x - 1} = \frac{1}{x - 1}, x=1x = 1 is extraneous.

Can it simplify fractions without an equals sign?

For simplifying, adding or dividing algebraic fractions without an equals sign, use the rational expression calculator — it cancels common factors and lists excluded values.