Solving Quadratic Equations: Complete IGCSE Guide | 二次方程求解：IGCSE 完整指南

📚 Solving Quadratic Equations: Complete IGCSE Guide | 二次方程求解：IGCSE 完整指南

Quadratic equations appear in almost every IGCSE Mathematics paper, whether you take the Core or the Extended route. You will be asked to solve them algebraically, interpret their graphs, or apply them to real-world problems such as area and projectile motion. This guide gives you a step-by-step system, with worked examples and exam tips, so you can handle any quadratic question with confidence.

二次方程几乎出现在每一份 IGCSE 数学试卷中，无论你参加 Core 还是 Extended 考试。你都会被要求以代数方法求解、解读二次函数的图像，或将其应用于面积、抛体运动等现实问题。本指南为你提供一套循序渐进的体系，配合典型例题与考试技巧，帮助你自信应对任何二次方程相关题目。

1. The Standard Form | 标准形式

A quadratic equation is a polynomial equation of degree 2, and its standard form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The value a is called the leading coefficient, b is the coefficient of x, and c is the constant term. If a were 0, the equation would become linear, not quadratic.

二次方程是次数为 2 的多项式方程，其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。a 称为首项系数，b 称为一次项系数，c 称为常数项。如果 a 为 0，方程就会变成一次方程，而非二次方程。

For example, in the equation 3x² – 5x + 2 = 0, we have a = 3, b = -5 and c = 2. In the exam, equations may be disguised: x² = 6x – 8, 2x(x + 1) = 5, or (x – 1)² = 4. You must always expand brackets and move every term to one side so the equation equals 0 before you begin solving.

例如，在方程 3x² – 5x + 2 = 0 中，a = 3，b = -5，c = 2。考试中方程可能被”伪装”起来：x² = 6x – 8、2x(x + 1) = 5 或 (x – 1)² = 4。你必须先展开括号，并将所有项移到一侧，使方程等于 0，然后再开始求解。

Example 1: Write 5x(2 – x) = 3x – 7 in standard form and state a, b and c.

例 1：将 5x(2 – x) = 3x – 7 化为标准形式，并指出 a、b、c。

Expanding gives 10x – 5x² = 3x – 7. Rearranging: -5x² + 10x – 3x + 7 = 0, so -5x² + 7x + 7 = 0. Multiplying by -1 gives 5x² – 7x – 7 = 0, so a = 5, b = -7, c = -7. Many examiners prefer the leading coefficient to be positive, so multiply by -1 whenever necessary.

展开得 10x – 5x² = 3x – 7。整理：-5x² + 10x – 3x + 7 = 0，即 -5x² + 7x + 7 = 0。两边乘以 -1 得到 5x² – 7x – 7 = 0，因此 a = 5，b = -7，c = -7。多数考官希望首项系数为正，所以需要时应将两边同乘 -1。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic has simple integer factors. The central idea is the zero product property: if p × q = 0, then either p = 0 or q = 0. So if you can rewrite ax² + bx + c as (mx + n)(px + q), then you can read off the two solutions directly.

当二次方程具有简单的整数因子时，因式分解是最快捷的方法。其核心是零乘积性质：若 p × q = 0，则 p = 0 或 q = 0。因此，若能将 ax² + bx + c 改写为 (mx + n)(px + q)，就可以直接读出两个解。

Example 2: Solve x² – 5x + 6 = 0 by factorisation.

例 2：用因式分解法解 x² – 5x + 6 = 0。

Look for two numbers that multiply to 6 and add to -5. They are -2 and -3, because (-2) × (-3) = 6 and (-2) + (-3) = -5. Hence x² – 5x + 6 = (x – 2)(x – 3) = 0, so x = 2 or x = 3.

找两个数，使它们的乘积为 6、和为 -5。这两个数是 -2 和 -3，因为 (-2) × (-3) = 6，且 (-2) + (-3) = -5。因此 x² – 5x + 6 = (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。

Watch for special patterns. A difference of two squares like x² – 9 = 0 factorises as (x – 3)(x + 3) = 0, giving x = ± 3. A perfect square

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