Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Every year, examiners reward students who can confidently switch between factorisation, completing the square, and the quadratic formula. This revision guide covers every skill you need, with carefully worked examples and the exact type of commentary that earns full marks on exam papers.

二次方程是 IGCSE 数学中考查频率最高的知识点之一。每年，考官都会给那些能熟练切换因式分解法、配方法和求根公式的学生高分。本复习指南涵盖了所有必备技能，配有精心编写的例题解析，以及考试中能帮助你拿到满分的解题思路讲解。

---

1. What Is a Quadratic Equation | 什么是二次方程

a, b and c are real numbers and a ≠ 0. The condition a ≠ 0 is essential: if a were zero, the equation would reduce to a linear equation, losing its quadratic nature entirely.

二次方程是未知数最高次数恰好为 2 的多项式方程。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 为实数，且 a ≠ 0。条件 a ≠ 0 至关重要：如果 a 等于零，方程就会退化为一次方程，完全失去二次的特性。

ecause its highest power is 3. On the IGCSE syllabus, you must be able to recognise quadratics quickly and know which method of solution is most appropriate for each situation.

例如，3x² − 2x + 5 = 0 是二次方程，而 3x − 2 = 0 不是。同样地，x³ − 4x = 0 是三次方程，因为它的最高次数是 3。在 IGCSE 考纲中，你必须能够快速识别二次方程，并判断在每种情况下哪种解法最合适。

---

2. Standard Form and Identifying Coefficients | 标准形式与系数识别

n exam error is forgetting the negative sign on b, which completely changes the solutions.

在求解任何二次方程之前，必须先写成标准形式：ax² + bx + c = 0。系数 a、b、c 必须包含其正负号。对于方程 2x² − 5x + 3 = 0，系数为 a = 2，b = −5，c = 3。一个常见的考试错误是忘记 b 的负号，这会完全改变解的结果。

is rearrangement step is one of the most common causes of lost marks in algebra questions.

许多方程一开始并非标准形式。以 3x² = 6x + 9 为例，要解这个方程，必须先重新整理：两边同时减去 6x 和 9，得到 3x² − 6x − 9 = 0。然后可以将每一项除以 3 进行化简，得到 x² − 2x − 3 = 0。省略重整步骤是代数题失分最常见的原因之一。

---

广告 / Sponsored

3. Solving by Factorisation | 因式分解法

Factorisation is generally the fastest method, provided the quadratic has simple integer factors. The principle is to write the quadratic as a product of two linear brackets. For a quadratic of the form x² + bx + c, find two numbers whose product is c and whose sum is b.

因式分解法通常是最快的方法，前提是二次式具有简单的整数因式。原理是将二次式写成两个一次括号的乘积。对于形式为 x² + bx + c 的二次式，找出两个数，使得它们的乘积为 c、和为 b。

Solve x² + 5x + 6 = 0

x + 2 = 0, giving x = −2, or x + 3 = 0, giving x = −3. The solutions are x = −2 or x = −3.

我们需要找两个数，它们的乘积为 6、和为 5。这两个数是 2 和 3，因为 2 × 3 = 6，2 + 3 = 5。因此方程变为 (x + 2)(x + 3) = 0。由于两个因式的乘积为零，至少有一个因式必须为零：要么 x + 2 = 0，得 x = −2；要么 x + 3 = 0，得 x = −3。所以解为 x = −2 或 x = −3。

ltiply to 6 and add to 7; these are 1 and 6. Split the middle term and factor by grouping:

当 x² 的系数不为 1 时，过程需要额外一步。求解 2x² + 7x + 3 = 0 时，先将 a 和 c 相乘：2 × 3 = 6。然后找两个数，使乘积为 6、和为 7；这两个数是 1 和 6。将中间项拆开，再用分组法因式分解：

2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3) = 0

Setting each bracket to zero gives 2x + 1 = 0, so x = −1/2, and

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）