Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in the IGCSE Edexcel Mathematics syllabus. They appear in almost every exam paper, both on the calculator and non-calculator papers, and they form the foundation for later topics such as graphs, inequalities, and calculus. In this article, we will explore three core methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula, together with the discriminant and practical word problems.

一元二次方程是 IGCSE Edexcel 数学考纲中最重要的话题之一。它几乎出现在每一份试卷中，无论是计算器试卷还是非计算器试卷，并且是后续学习函数图像、不等式以及微积分的基础。在本文中，我们将学习解一元二次方程的三种核心方法：因式分解法、配方法和公式法，同时还会介绍判别式以及实际应用题。

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1. What is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is a polynomial equation of degree 2, meaning that the highest power of the unknown variable is 2. The standard form of a quadratic equation is:

一元二次方程是最高次数为 2 的多项式方程，也就是说未知数的最高次幂为 2。一元二次方程的标准形式为：

ax² + bx + c = 0

where a, b and c are constants, and a ≠ 0. If a = 0, the equation becomes linear rather than quadratic. For example, 2x² − 5x + 3 = 0 is a quadratic equation with a = 2, b = −5, and c = 3.

其中 a、b、c 为常数，且 a ≠ 0。如果 a = 0，方程就变成了一次方程而不是二次方程。例如，2x² − 5x + 3 = 0 就是一个一元二次方程，其中 a = 2，b = −5，c = 3。

• Key point: the coefficient a can never be zero, otherwise the equation is not quadratic.

  关键点：系数 a 绝不能为零，否则方程就不再是二次方程。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer roots. The method relies on rewriting ax² + bx + c as a product of two brackets. When a = 1, we look for two numbers whose product is c and whose sum is b.

当一元二次方程有简单的整数根时，因式分解法通常是最快捷的方法。该方法的核心是把 ax² + bx + c 改写为两个括号的乘积。当 a = 1 时，我们要找两个数，使它们的乘积等于 c，并且它们的和等于 b。

Example: Solve x² + 5x + 6 = 0.

例题：解 x² + 5x + 6 = 0。

We need two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3, because 2 × 3 = 6 and 2 + 3 = 5. Therefore:

我们需要找到两个数，使它们的乘积为 6，并且和为 5。这两个数是 2 和 3，因为 2 × 3 = 6，且 2 + 3 = 5。因此：

(x + 2)(x + 3) = 0

For the product of two brackets to equal zero, at least one bracket must be zero. Hence x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

要使两个括号的乘积等于零，至少有一个括号必须为零。因此 x + 2 = 0 或 x + 3 = 0，解得 x = −2 或 x = −3。

When a ≠ 1, factorisation requires a little more care. For example, to solve 2x² + 7x + 3 = 0, we first multiply a by c to get 2 × 3 = 6, then look for factor pairs of 6 that add to 7. The pair is 1 and 6, so we split the middle term:

当 a ≠ 1 时，因式分解需要更加细心。例如，解 2x² + 7x + 3 = 0 时，我们先把 a 和 c 相乘得到 2 × 3 = 6，然后找出和为 7 的因数对。这一对是 1 和 6，所以我们拆分中间项：

2x² + x + 6x + 3 = 0

Then we factor by grouping: x(2x + 1) + 3(2x + 1) = 0, which gives (x + 3)(2x + 1) = 0. Hence x = −3 or x = −½.

然后分组因式分解：x(2x + 1) + 3(2x + 1) = 0，得到 (x + 3)(2x + 1) = 0。因此 x = −3 或 x = −½。

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3. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic equation, even when factorisation is not possible. The idea is to rewrite the equation in the form (x + p)² = q, and then take square roots of both sides.

配方法是一种非常强大的技巧，适用于任何一元二次方程，即使无法因式分解也能使用。其核心思想是把方程改写为 (x + p)² = q 的形式，然后对方程两边同时开平方根。

When a = 1, the quadratic x² + bx + c can be rewritten as:

当 a = 1 时，二次式 x² + bx + c 可以改写为：

x² + bx + c = (x + b/2)² − (b/2)² + c

Example: Solve x² + 6x − 7 = 0 by completing the square.

例题：用配方法解 x² + 6x − 7 = 0。

Take half of 6, which is 3. Then:

取 6 的一半，即 3。然后：

(x + 3)² − 9 − 7 = 0

Simplifying gives (x + 3)² = 16. Taking square roots gives x + 3 = ±4, hence x = 1 or x = −7.

化简得 (x + 3)² = 16。两边开平方根得 x + 3 = ±4，因此 x = 1 或 x = −7。

If the coefficient of x² is not 1, first divide every term by a. For example, 2x² + 8x + 5 = 0 becomes x² + 4x + 2.5 = 0 before completing the square.

如果 x² 的系数不是 1，先将每一项都除以 a。例如，2x² + 8x + 5 = 0 在配方之前先变为 x² + 4x + 2.5 = 0。

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4. Solving by the Quadratic Formula | 公式法

The quadratic formula can solve any quadratic equation and is especially useful when the roots are not integers or simple fractions. Given ax² + bx + c = 0, the roots are given by:

公式法可以求解任何一元二次方程，当根不是整数或简单分数时特别有用。对于 ax² + bx + c = 0，求根公式为：

x = (−b ± √(b² − 4ac)) / (2a)

This formula is printed in the Edexcel formula booklet, so you do not need to memorise it, but you must be comfortable substituting values correctly. Be especially careful with signs when b is negative.

这个公式印在 Edexcel 的公式册中，你不需要背诵，但必须能够正确代入数值。当 b 为负数时，要特别注意符号问题。

Example: Solve 2x² − 4x −

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