Problems Leading to Quadratic Equations | 二次方程的应用问题

📚 Problems Leading to Quadratic Equations | 二次方程的应用问题

Quadratic equations appear frequently in the IGCSE Edexcel Mathematics syllabus, not only as abstract exercises but also as models for real-life problems. These "word problems" require you to read a scenario, identify an unknown quantity, and set up a quadratic equation that represents the given information. Mastering these problems improves your algebraic thinking and problem-solving skills, and they are common in exam papers.

在IGCSE Edexcel数学大纲中，二次方程不仅以抽象练习形式出现，还常作为实际问题的模型。这些“文字题”要求你阅读情境、识别未知量，并建立一个表示给定信息的二次方程。掌握这类问题能提高你的代数思维和解题能力，它们在考试中也很常见。

1. What Are Quadratic Word Problems? | 什么是二次方程文字题？

0. These problems are not just about solving the equation; they test your ability to model real-world scenarios mathematically.

二次方程文字题是对一种情境的描述，将其转化为代数形式后，会得到一个形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。这些问题不仅仅是解方程，它们考察的是你将实际情境数学建模的能力。

Typical topics include finding dimensions of a rectangle given its area and perimeter, determining two consecutive numbers with a certain property, or calculating the time taken for a moving object to reach a specific position.

典型主题包括：给定面积和周长求矩形的长宽、确定满足某种性质的两个连续数，或者计算运动物体到达特定位置所需的时间。

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2. Key Steps to Solve Quadratic Word Problems | 解二次方程文字题的关键步骤

To tackle any problem leading to a quadratic equation, follow these systematic steps:

要解决任何导出二次方程的问题，请遵循以下系统步骤：

Step 1: Read the problem carefully and understand what quantity you need to find.

第一步：仔细阅读问题，明确你需要求的是什么量。

Step 2: Choose a variable (often x) to represent the unknown quantity. If there are two unknowns, express both in terms of the same variable.

第二步：选择一个变量（通常用 x）表示未知量。如果有两个未知数，用同一个变量表示它们。

Step 3: Write an equation based on the relationships described in the problem. This will involve areas, products, sums, or the Pythagorean theorem, leading to a quadratic form.

第三步：根据问题中描述的关系写出方程。这可能涉及面积、乘积、求和或勾股定理，从而得到一个二次方程。

Step 4: c = 0 and solve it by factorising, completing the square, or using the quadratic formula.

第四步：将方程整理为标准形式 ax² + bx + c = 0，然后通过因式分解、配方法或求根公式求解。

Step 5: Interpret the solutions in the context of the problem. Discard any solution that does not make sense (e.g., negative length or time).

第五步：结合问题情境解释解。舍弃任何不合理的解（例如负的长度或时间）。

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3. Forming Equations: Translating Words into Algebra | 建立方程：将文字转化为代数式

ften translating the English description into mathematical expressions. Look for keywords: "product" means multiply, "sum" means add, "difference" means subtract, "square of" means raise to power 2, "area" often leads to length × width, and "a is twice b" becomes a = 2b.

最具挑战性的部分往往是将英文描述转化为数学表达式。寻找关键词：“乘积”表示相乘，“和”表示相加，“差”表示相减，“……的平方”表示指数为2，“面积”常常导出长 × 宽，“a 是 b 的两倍”变为 a = 2b。

Example: "The product of two consecutive integers is 72." first integer is x, the next consecutive integer is x + 1. The equation is x(x + 1) = 72.

示例：“两个连续整数的乘积为72。” 设第一个整数为 x，则下一个连续整数为 x + 1。方程为 x(x + 1) = 72。

Example: "The length of a rectangle is 3 cm more than its width, and the area is 40 cm²." Let width be w, then length = w + 3. Equation: w(w + 3) = 40.

示例：“一个矩形的长比宽多3 cm，面积为40 cm²。” 设宽为 w，则长 = w + 3。方程：w(w + 3) = 40。

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4. Solving Quadratic Equations by Factorising | 通过因式分解解二次方程

Once the equation is in standard form, factorising is often the quickest method if the quadratic is factorisable. Remember to set each factor equal to zero.

一旦方程化为标准形式，如果二次式可分解，因式分解通常是最快的方法。记住令每个因式等于零。

Example: x² - 5x + 6 = 0 factorises to (x - 2)(x - 3) = 0, so x = 2 or x = 3.

示例：x² - 5x + 6 = 0 分解为 (x - 2)(x - 3) = 0，因此 x = 2 或 x = 3。

For instance, 2x² + 5x - 3 = 0 factorises to (2x - 1)(x + 3) = 0, giving x = ½ or x = -3.

当 x² 的系数不是1时，需要仔细分解。例如，2x² + 5x - 3 = 0 分解为 (2x - 1)(x + 3) = 0，得出 x = ½ 或 x = -3。

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5. Using the Quadratic Formula | 使用求根公式

If the quadratic does not factorise easily, apply the quadratic formula:

如果二次式不易分解，则使用求根公式：

x = [-b ± √b² - 4ac] / (2a)

y a, b, and c correctly, including their signs. The discriminant b² - 4ac tells you about the nature of the roots.

此公式可解任何二次方程 ax² + bx + c = 0。你必须正确识别 a、b 和 c，包括它们的符号。判别式 b² - 4ac 告诉你根的性质。

For word problems, you often get two mathematical solutions but only one fits the real-life context.

对于文字题，你常常会得到两个数学解，但只有一个符合实际情境。

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6. Area and Dimension Problems | 面积与尺寸问题

These are the most frequent quadratic word problems. A typical question: "The length of a rectangular garden is 4 m more than its width. The area is 60 m². Find the dimensions."

这是最常见的二次方程文字题。一个典型问题：“一个矩形花园的长比宽多4米，面积为60平方米。求尺寸。”

. Solutions: x = -10 or x = 6. Since width cannot be negative, width = 6 m, length = 10 m.

设宽为 x 米，则长 = x + 4 米。面积 = x(x + 4) = 60。展开得 x² + 4x - 60 = 0。因式分解：(x + 10)(x - 6) = 0。解为 x = -10 或 x = 6。因为宽度不能为负，所以宽 = 6 米，长 = 10 米。

Sometimes the problem involves a border or a path, increasing the dimensions. Always draw a diagram to visualise the situation.

有时问题涉及边界或道路，增加了尺寸。一定要画图来形象化情境。

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7. Problems with Consecutive Numbers | 连续数问题

Consecutive integers, consecutive even numbers, or consecutive odd numbers are common. For consecutive even numbers, define the first as n, then the next is n + 2, not n + 1.

连续整数、连续偶数或连续奇数都很常见。对于连续偶数，设第一个为 n，则下一个是 n + 2，而不是 n + 1。

Example: "The product of two consecutive even numbers is 48. Find the numbers." airs are (-8, -6) and (6, 8). Both are valid because the problem does not specify positive numbers.

示例：“两个连续偶数的乘积为48。求这两个数。”设较小的为 x，则下一个为 x + 2。方程：x(x + 2) = 48 → x² +

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