Solving Quadratic Equations | 一元二次方程

📚 Solving Quadratic Equations | 一元二次方程

Quadratic equations are among the most frequently tested topics in IGCSE Mathematics. They appear in both Paper 2 and Paper 4, either as standalone questions or embedded in geometry, statistics and coordinate geometry contexts. This revision guide covers every method you need, with worked examples and a clear explanation of the most common pitfalls.

一元二次方程是 IGCSE 数学中考察频率最高的考点之一。它既会作为独立题目出现在 Paper 2 和 Paper 4 中，也常与几何、统计和坐标几何背景相结合。本复习指南涵盖你所需要的全部解法，并配有完整例题及对常见陷阱的清晰讲解。

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

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 the variable x is 2, which is what makes it “quadratic” rather than linear.

一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，这正是它区别于一次方程（线性方程）的关键。

There are three core solution methods that you must master for the exam:

考试中你必须掌握三种核心解法：

• Factorisation – write the quadratic as a product of two brackets, then set each bracket to zero.

  因式分解法——将二次式写成两个括号的乘积，再令每个括号等于零。

• Completing the square – rewrite the equation in the form (x + p)² = q, then take square roots.

  配方法——将方程改写为 (x + p)² = q 的形式，再两边开平方。

• Quadratic formula – substitute a, b and c directly into x = (-b ± √(b² – 4ac)) / 2a.

  求根公式——直接把 a、b、c 代入 x = (-b ± √(b² – 4ac)) / 2a。

Knowing when to use each method is a skill in itself. Factorisation is fastest for simple integer roots; completing the square is best for exact roots and turning points; the formula always works, making it your safety net.

知道何时选择哪种方法本身就是一项技能。因式分解法在根为简单整数时最快；配方法适合求精确根和顶点；求根公式则永远适用，是你的”安全网”。

2. Standard Form and Rearranging | 标准形式与整理

Before solving, you must rearrange the equation into the standard form ax² + bx + c = 0. This means moving every term to one side of the equals sign so that the other side is exactly zero.

求解之前，必须先将方程整理成标准形式 ax² + bx + c = 0，即把所有项移到等号的一边，使另一边恰好为零。

Example: x² + 5x = 14 → x² + 5x – 14 = 0

示例：x² + 5x = 14 → x² + 5x – 14 = 0

Always check that the x² term is positive and that the equation is fully simplified before you begin. If the coefficient of x² is not 1, you keep it when applying the formula, but you may divide the whole equation by a common factor if every coefficient shares one.

开始之前务必检查 x² 项系数为正，并确认方程已完全化简。如果 x² 的系数不是 1，套用求根公式时应保留；但若所有系数都有公因数，可以先对整个方程除以该公因数。

Sometimes the quadratic appears hidden inside fractions. For example, 3/x + x = 5 can be multiplied by x to give 3 + x² = 5x, then rearranged to x² – 5x + 3 = 0. Multiplying by the denominator is a standard first step whenever x appears on the denominator.

有时二次方程会隐藏在分式中。例如，3/x + x = 5 两边同乘 x 得 3 + x² = 5x，再整理为 x² – 5x + 3 = 0。只要 x 出现在分母中，两边同乘分母就是标准的第一步。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer factors. The key idea is to find two numbers that multiply to give ac and add to give b.

当二次方程具有简单的整数因子时，因式分解是最快的方法。核心思路是找到两个数，使它们的乘积等于 ac，和等于 b。

Follow these steps for ax² + bx + c =

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