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Solving Algebraic Fractional Equations | Edexcel IGCSE Maths | 含代数分式方程的解法

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📚 Solving Algebraic Fractional Equations | 含代数分式方程的解法

Fractional equations contain algebraic fractions – expressions with a variable in the numerator or denominator. Solving them is a key skill for Edexcel IGCSE Mathematics, combining algebraic manipulation with careful checks for undefined values.

分式方程是指含有代数分式（分子或分母中带有变量）的方程。解这类方程是 Edexcel IGCSE 数学的核心技能，需要将代数变形与对无定义值的仔细检查结合起来。

1. What are Fractional Equations? | 什么是分式方程

A fractional equation is an equation in which at least one term is an algebraic fraction, such as \(\frac{3}{x+2}\) or \(\frac{2x-1}{x}\). These equations often arise from ratios, rates, or geometric problems.

分式方程是至少含有一个代数分式项（如 \(\frac{3}{x+2}\) 或 \(\frac{2x-1}{x}\)）的方程。这类方程常出现在比例、速率或几何问题中。

Example:

例如：

\(\frac{5}{x} + \frac{1}{x-2} = 3\)

Here the unknown \(x\) appears in the denominator, so we must be careful about values that make a denominator zero.

这里未知数 \(x\) 出现在分母中，因此必须注意使分母为零的值。

2. Domain and Undefined Values | 定义域与无定义值

Before solving, identify all values of the variable that make any denominator equal to zero. These values are not allowed in the solution set.

在求解之前，先找出使任意分母等于零的变量值。这些值不允许出现在解集中。

For example, in the equation:

例如，在方程中：

\(\frac{2}{x-3} – \frac{1}{x+1} = 4\)

the denominators are \(x-3\) and \(x+1\). So \(x \neq 3\) and \(x \neq -1\).

分母为 \(x-3\) 和 \(x+1\)，因此 \(x \neq 3\) 且 \(x \neq -1\)。

Steps to find excluded values:

求排除值的步骤：

  • Set each denominator equal to zero.

    令每个分母等于零。

  • Solve the resulting linear or quadratic equations.

    解所得的线性或二次方程。

  • Exclude those values from the final answer.

    将这些值从最终答案中排除。

3. Eliminating Denominators | 去分母

The main strategy is to multiply both sides of the equation by the lowest common multiple (LCM) of all denominators. This clears the fractions and produces a simpler equation.

主要策略是对方程两边同时乘以所有分母的最小公倍数（LCM），从而消去分母，得到更简单的方程。

For the equation:

对于方程：

\(\frac{2}{x} + \frac{3}{x+1} = 5\)

the LCM is \(x(x+1)\). Multiply every term by this LCM:

最小公倍数为 \(x(x+1)\)。将每一项都乘以这个 LCM：

\(2(x+1) + 3x = 5x(x+1)\)

Now expand and solve the resulting equation.

然后展开并解所得方程。

4. Solving Linear Fractional Equations | 解线性分式方程

After clearing denominators, if the resulting equation is linear, solve it by isolating the variable.

去分母后，如果所得方程是线性的，则通过移项合并来求解变量。

Worked example:

示例：

\(\frac{3}{x-2} = \frac{2}{x+1}\)

Multiply both sides by \((x-2)(x+1)\):

两边同时乘以 \((x-2)(x+1)\)：

\(3(x+1) = 2(x-2)\)

\(3x + 3 = 2x – 4\)

\(x = -7\)

Check that \(x \neq 2\) and \(x \neq -1\), so \(x = -7\) is valid.

检查 \(x \neq 2\) 且 \(x \neq -1\)，因此 \(x = -7\) 有效。

5. Solving Quadratic Fractional Equations | 解二次分式方程

Often clearing denominators produces a quadratic equation. Solve it by factorising, completing the square, or using the quadratic formula.

去分母后常常会得到二次方程。可以通过因式分解、配方法或求根公式来解。

Example:

例题：

\(\frac{4}{x} – \frac{3}{x+2} = 1\)

LCM is \(x(x+2)\). Multiply all terms:

最小公倍数为 \(x(x+2)\)。各项乘以它：

\(4(x+2) – 3x = x(x+2)\)

\(4x + 8 – 3x = x^2 + 2x\)

\(x + 8 = x^2 + 2x\)

\(x^2 + x – 8 = 0\)

This does not factorise nicely, so use the quadratic formula:

此式不易因式分解，因此使用求根公式：

\(x = \frac{-1 \pm \sqrt{1 + 32}}{2} = \frac{-1 \pm \sqrt{33}}{2}\)

Both values are valid because neither equals 0 or -2.

两个值均有效，因为它们都不等于 0 或 -2。

6. Checking Solutions and Extraneous Roots | 验根与增根

Multiplying by an expression containing a variable can introduce extraneous roots – values that satisfy the transformed equation but not the original one.

乘以含变量的表达式可能会引入增根——即满足变形方程但不满足原方程的值。

Always check each candidate solution by substituting into the original denominators:

务必通过代入原分母来检验每个候选解：

  • Substitute the value into every denominator of the original equation.

    将该值代入原方程的每一个分母。

  • If any denominator becomes zero, discard that value.

    如果任何一个分母为零，则舍去该值。

  • If all denominators are non-zero, verify that the original equation holds true.

    如果所有分母都不为零，再验证原方程是否成立。

Example of an extraneous root:

增根示例：

\(\frac{x^2}{x-1} – \frac{1}{x-1} = 2\)

Combine the fractions:

合并分式：

\(\frac{x^2 – 1}{x-1} = 2\)

\(x^2 – 1 = 2(x-1)\)

\(x^2 – 2x + 1 = 0\)

\((x-1)^2 = 0\)

So \(x = 1\) appears as a solution. But \(x = 1\) makes the original denominator \(x-1 = 0\), so it is not allowed. The equation has no solution.

于是 \(x = 1\) 看似是解，但 \(x = 1\) 使原分母 \(x-1 = 0\)，因此不允许。该方程无解。

7. Common Pitfalls | 常见错误

Students often make mistakes when dealing with algebraic fractions. Avoid these common pitfalls:

学生在处理代数分式时经常出错。避免以下常见错误：

  • Forgetting to exclude values that make denominators zero.

    忘记排除使分母为零的值。

  • Multiplying only the visible fractions by the LCM, not every term on both sides.

    只将“明显的分数”乘以 LCM，而不是两边每一项都乘。

  • Sign errors when subtracting a fraction with multiple terms in the numerator.

    当分子含有多项式并做减法时，符号出错。

  • Not checking for extraneous roots at the end.

    最后不检验增根。

  • Expanding incorrectly when simplifying products like \((x+1)(x-2)\).

    展开如 \((x+1)(x-2)\) 等乘积时出错。

Always write down the excluded values before you begin, and check all answers.

建议开始前先写出排除值，并检查所有答案。

8. Worked Example (Exam-Style) | 考试题型例题

Solve the equation, giving your answer correct to 2 decimal places where necessary:

解下列方程，必要时将答案精确到小数点后两位：

\(\frac{2}{x+3} + \frac{5}{x-2} = 1\)

Step 1: Excluded values: \(x \neq -3\), \(x \neq 2\).

第一步：排除值：\(x \neq -3\)，\(x \neq 2\)。

Step 2: Multiply both sides by the LCM \((x+3)(x-2)\):

第二步：两边同时乘以 LCM \((x+3)(x-2)\)：

\(2(x-2) + 5(x+3) = (x+3)(x-2)\)

Step 3: Expand both sides:

第三步：展开两边：

\(2x – 4 + 5x + 15 = x^2 + x – 6\)

\(7x + 11 = x^2 + x – 6\)

Step 4: Rearrange to form a quadratic equation:

第四步：移项得二次方程：

\(x^2 – 6x – 17 = 0\)

Step 5: Use the quadratic formula with \(a=1\), \(b=-6\), \(c=-17\):

第五步：用求根公式，其中 \(a=1\)，\(b=-6\)，\(c=-17\)：

\(x = \frac{6 \pm \sqrt{36 + 68}}{2} = \frac{6 \pm \sqrt{104}}{2}\)

\(x = \frac{6 \pm 2\sqrt{26}}{2} = 3 \pm \sqrt{26}\)

Step 6: Check both values: \(3 + \sqrt{26} \approx 8.10\) and \(3 – \sqrt{26} \approx -2.10\). Neither equals -3 or 2, so both are valid.

第六步：检验两个值：\(3 + \sqrt{26} \approx 8.10\)，\(3 – \sqrt{26} \approx -2.10\)。都不等于 -3 或 2，因此两者均有效。

9. Practice Questions | 练习

Try these questions to build confidence. Solve each equation and state any excluded values.

尝试以下题目以增强信心。解每个方程并写出排除值。

  1. \(\frac{4}{x} = \frac{3}{x-1}\)

    \(\frac{4}{x} = \frac{3}{x-1}\)

  2. \(\frac{1}{x} + \frac{2}{x+2} = 1\)

    \(\frac{1}{x} + \frac{2}{x+2} = 1\)

  3. \(\frac{x}{x-2} – \frac{2}{x+3} = \frac{1}{x-2}\)

    \(\frac{x}{x-2} – \frac{2}{x+3} = \frac{1}{x-2}\)

  4. \(\frac{3}{x^2-1} + \frac{2}{x+1} = 1\)

    \(\frac{3}{x^2-1} + \frac{2}{x+1} = 1\)

Answers:

答案：

  1. \(x = 4\)

    \(x = 4\)

  2. \(x = 1 \pm \sqrt{3}\) (both valid)

    \(x = 1 \pm \sqrt{3}\)（均有效）

  3. No solution (the candidate \(x=-3\) makes denominators zero)

    无解（候选值 \(x=-3\) 使分母为零）

  4. \(x = -4\) or \(x = 1\) (check: \(x=1\) is valid because \(x^2-1=0\)? Actually \(1^2-1=0\), so \(x=1\) is excluded; only \(x=-4\) is valid)

    \(x = -4\) 或 \(x = 1\)（注意：\(x=1\) 时 \(1^2-1=0\)，因此 \(x=1\) 被排除；只有 \(x=-4\) 有效）

For question 4, let’s verify: original \(\frac{3}{x^2-1} + \frac{2}{x+1} = 1\). Factor \(x^2-1=(x-1)(x+1)\). LCM is \((x-1)(x+1)\). Multiply: \(3 + 2(x-1) = (x^2-1)\). So \(3+2x-2 = x^2-1\) → \(2x+1 = x^2-1\) → \(x^2-2x-2=0\)? Wait, that gives different answer. Let me recalc. Actually \(3+2(x-1) = (x-1)(x+1)\) → \(3+2x-2 = x^2-1\) → \(2x+1 = x^2-1\) → \(x^2 – 2x – 2 = 0\). This doesn’t factor nicely. So my previous answer is wrong. Need to correct.

Let’s properly solve question 4:
Equation: 3/(x^2-1) + 2/(x+1) = 1.
Denominators: x^2-1 = (x-1)(x+1) and x+1. LCM = (x-1)(x+1).
Multiply each term by LCM:
3 + 2(x-1) = (x-1)(x+1)
3 + 2x – 2 = x^2 – 1
2x + 1 = x^2 – 1
x^2 – 2x – 2 = 0
Use quadratic formula: x = [2 ± sqrt(4+8)]/2 = [2 ± sqrt(12)]/2 = [2 ± 2√3]/2 = 1 ± √3.
Excluded values: x^2-1=0 → x=±1; x+1=0 → x=-1. So x=1 or -1 excluded. 1±√3 are approximately 2.732 and -0.732, neither excluded. So valid solutions are x = 1 ± √3.

Question 3: x/(x-2) – 2/(x+3) = 1/(x-2). LCM = (x-2)(x+3). Multiply:
x(x+3) – 2(x-2) = 1*(x+3)
x^2+3x -2x+4 = x+3
x^2+x+4 = x+3
x^2+1 = 0 → no real solutions. So no solution. Excluded values x=2, x=-3. The previous answer “candidate x=-3 makes denominators zero” was nonsense. Need proper.

Question 2: 1/x + 2/(x+2) = 1. LCM x(x+2). Multiply: (x+2) + 2x = x(x+2) → 3x+2 = x^2+2x → x^2 – x -2 = 0 → (x-2)(x+1)=0 → x=2 or x=-1. Excluded x=0, x=-2. Both valid. But my earlier answer “1±√3” is wrong. I need to redo the practice answers correctly.

Let’s solve question 2 properly:
1/x + 2/(x+2) = 1. LCM x(x+2). Multiply each term:
(x+2) + 2x = x(x+2)
3x+2 = x^2+2x
x^2 – x – 2 = 0
(x-2)(x+1)=0 → x=2, x=-1. Valid because x≠0, x≠-2. So answers x=2, x=-1.

Question 1: 4/x = 3/(x-1). Cross multiply: 4(x-1)=3x → 4x-4=3x → x=4. Valid since x≠0, x≠1.

So practice answers:
1. x=4
2. x=2 or x=-1
3. No real solutions (since x^2+1=0 has no real roots)
4. x = 1 ± √3

But in the script I wrote “answer for 4: x=-4 or x=1” which is wrong. Need to correct the article. I’ll revise that section.

Also in question 3, note that x/(x-2) – 2/(x+3) = 1/(x-2). The term 1/(x-2) has denominator x-2, LCM (x-2)(x+3). Multiplication: x(x+3) – 2(x-2) = 1*(x+3). That’s correct. Then x^2+3x-2x+4 = x+3 → x^2+x+4 = x+3 → x^2+1=0. No real solutions. So indeed no solution.

I need to update the practice section accordingly. Also ensure the formatting has

    and
  1. with pairs of

    ? In the spec says “每个要点先写英文

    ，紧跟写中文

    配对” for lists use

    • . For ordered list, likely use
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          similarly. Better to have each
        1. with two

          s. Or just one

          with both languages? Spec says “每个要点先写英文

          ，紧跟写中文

          配对” so inside each

        2. I need two

          elements. I’ll do that.

          Let’s restructure practice questions with correct answers and explanations. Also ensure the “Practice Questions” section is not too long, but okay.

          Now, I need to ensure no LaTeX. I have been using \( \) and \[ \] in the HTML, which is LaTeX. The instruction says strictly no LaTeX, like \( \), \[ \], $$, \ce{}, \frac. Actually, using \( \) is LaTeX. I must avoid that. Instead, I should use Unicode characters and maybe HTML tags. How to represent variables and equations? We can use plain text with italics? Since HTML allows or , but formulas should be displayed as centered bold using Unicode. The instruction says “公式/方程用

          …

          居中加粗；数学/化学符号一律用 Unicode（上标 ⁻³ ² ⁺、下标 ₐ ₙ ₂、→ ⇌ × ÷ ± ½ √ ∫ Δ Σ π θ ° 等）；严禁使用 LaTeX，如 \( \)、\[ \]、$$、\ce{}、\frac 一律不用”.

          So I must not use backslash parentheses or backslash brackets. I need to represent fractions using Unicode? For example, “3/(x+2)” or use “3⁄(x+2)”? The instruction gives examples of Unicode symbols but doesn’t specify fraction slash. However, using “/” in plain text is acceptable? LaTeX \frac is forbidden, but I can use “3/(x+2)” or “3⁄x” maybe. To be safe, use “/” for fractions, and use superscripts for powers, e.g., x² for x^2, x²+1 for x^2+1. For subscripts, use Unicode subscript numbers if available, e.g., x₁, x₂. For square root, use √ symbol. For plusminus use ±. For multiplication use ×.

          Also “Solving Algebraic Fractional Equations” might contain “Algebraic” etc. The title in TITLE line: “Solving Algebraic Fractional Equations | Edexcel IGCSE Maths | 含代数分式方程的解法” Actually the format says: TITLE: | . So I should put English first then Chinese, separated by ” | “. So “TITLE: Solving Algebraic Fractional Equations | Edexcel IGCSE Maths | 含代数分式方程的解法”? Wait no, the pattern is “TITLE: | ” So it should be “TITLE: Solving Algebraic Fractional Equations | 含代数分式方程的解法” But the subject? Actually the title might be “Edexcel IGCSE Maths: Solving Algebraic Fractional Equations” but the given subject is “数学” and topic. The instruction says “第一行只输出： TITLE: | ” So English title can be “Solving Algebraic Fractional Equations | Edexcel IGCSE Maths” or “Edexcel IGCSE Maths: Solving Algebraic Fractional Equations”. But it must have English and Chinese separated by ” | “. So perhaps English: “Solving Algebraic Fractional Equations (Edexcel IGCSE Maths)” then Chinese: “含代数分式方程的解法”. But the example title in the opening h2 says “📚 | ” So same. So I’ll choose English Title as “Solving Algebraic Fractional Equations | Edexcel IGCSE Maths” actually the “|” inside English title might be confusing. Better to follow exactly: English Title should be “Solving Algebraic Fractional Equations for Edexcel IGCSE Mathematics” or simpler “Solving Algebraic Fractional Equations”. Then Chinese “含代数分式方程的解法”. So TITLE line: “TITLE: Solving Algebraic Fractional Equations | 含代数分式方程的解法”. The subject can be inferred from context. I’ll do that.

          Now, in the body, I need to replace all LaTeX with Unicode. For example, fractions: use “3/(x+2)” instead of \frac{3}{x+2}. But to make it look like a fraction, could use “3⁄(x+2)”? Unicode fraction slash U+2044. But “3/(x+2)” is fine. For variables, use x perhaps? But the spec says “数学/化学符号一律用 Unicode” not necessarily HTML tags. We can use or just italic via CSS? It says “一律用 Unicode” meaning use Unicode characters for symbols. For variables, we can use plain “x”, “y”. I’ll use plain text.

          For powers: x², x³, x⁴, etc. For subscripts: x₁, x₂, but rarely needed. For square root: √ (U+221A). For plus-minus: ±. For multiplication: ×. For division: ÷. For fraction slash: ⁄ (U+2044) if needed, but I’ll just use “/”. For centered formulas, I’ll use tags.

          Also ensure not to use \left, \right, etc. So I’ll rewrite all equations in Unicode format.

          Let’s rewrite the entire article content with Unicode equations.

          I’ll start from TITLE and then the h2. Then the introduction. Then sections.

          I need to be careful with the word count. The article should be 3000-5000 Chinese+English characters. We’ll produce moderately long content.

          Let’s draft each section:

          – Introduction: two paragraphs.

          – Section 1: What are Fractional Equations? Definitions with examples.

          – Section 2: Domain and undefined values. Example.

          – Section 3: Eliminating denominators.

          – Section 4: Solving linear fractional equations.

          – Section 5: Solving quadratic fractional equations.

          – Section 6: Checking solutions and extraneous roots.

          – Section 7: Common pitfalls.

          – Section 8: Worked example.

          – Section 9: Practice questions.

          – Section 10: Summary.

          That’s 10 sections, within 8-12.

          Now write the body.

          Important: For each list, if using

          • , inside
          • put two

            elements. For ordered list, use

            1. similarly. The spec says “列表用
              • ；可用
                ” but doesn’t mention
                  . I’ll use
                  for bullet lists and
                  for numbered practice questions, but I’ll align with spec by using
                  where possible. For practice questions, I can use
                    with numbers in the text, or
                  . But to be safe, use