Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in nearly every IGCSE Mathematics paper. Mastering this topic unlocks higher marks across algebra, coordinate geometry, and problem-solving questions. This revision guide covers the key solution methods, the discriminant, graphing, and common exam traps.

一元二次方程几乎是每份 IGCSE 数学试卷的必考内容。掌握好这一专题，能帮助你在代数、坐标几何和应用题中获得更高分数。本复习指南涵盖主要解法、判别式、图像特征以及常见考试陷阱。

1. The Standard Form | 标准形式

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which is why it is called a ‘quadratic’.

一元二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。方程中 x 的最高次数为 2，因此称为二次方程。

If a = 0, the equation becomes linear, not quadratic. Always rearrange the equation into the form ax² + bx + c = 0 before applying any solution method.

如果 a = 0，方程就退化为一次方程，不再是二次方程。在使用任何解法之前，务必先将方程整理成 ax² + bx + c = 0 的形式。

2. Expanding and Factorising | 展开与因式分解

Before solving a quadratic, you must be fluent in expanding and factorising. The three most useful patterns are listed below.

在求解二次方程之前，你必须熟练展开与因式分解。下面列出三种最常用的模式。

• Common factor: 2x² + 6x = 2x(x + 3)

  提取公因式：2x² + 6x = 2x(x + 3)

• Difference of two squares: x² – 9 = (x + 3)(x – 3)

  平方差公式：x² – 9 = (x + 3)(x – 3)

• Perfect square: x² + 10x + 25 = (x + 5)²

  完全平方：x² + 10x + 25 = (x + 5)²

For a general trinomial x² + bx + c, find two numbers that multiply to c and add to b. For example, x² – 5x + 6 = (x – 2)(x – 3) because (-2) × (-3) = 6 and (-2) + (-3) = -5.

对于一般三项式 x² + bx + c，找到两个数，使其乘积等于 c、和等于 b。例如，x² – 5x + 6 = (x – 2)(x – 3)，因为 (-2) × (-3) = 6 且 (-2) + (-3) = -5。

When the coefficient of x² is not 1, use the ac method: multiply a and c, find two numbers with that product and sum b, split the middle term, then factor by grouping.

当 x² 的系数不为 1 时，可选用 ac 方法：先计算 a 与 c 的乘积，找到两个数使其乘积等于 ac、和等于 b，再拆分中间项并分组因式分解。

3. Solving by Factorising | 因式分解法求解

If a product of two factors equals zero, then at least one factor must be zero. This is the zero product property: if (x – m)(x – n) = 0, then x = m or x = n.

若两个因式的乘积为零，则至少有一个因式为零。这就是零乘积性质：若 (x – m)(x – n) = 0，则 x = m 或 x = n。

Example: Solve x² – 7x + 10 = 0.

示例：解方程 x² – 7x + 10 = 0。

Factorise: (x – 2)(x – 5) = 0. Therefore x – 2 = 0 or x – 5 = 0

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