Solving Quadratic Equations: Factorisation, Formula and Completing the Square | 解二次方程：因式分解、求根公式与配方法

📚 Solving Quadratic Equations: Factorisation, Formula and Completing the Square | 解二次方程：因式分解、求根公式与配方法

Quadratic equations are one of the most heavily tested topics in IGCSE Mathematics. You will meet them in both Paper 2 and Paper 4, in calculator and non-calculator forms. This article covers the three key solution methods, the discriminant, graph sketching, algebraic intersection problems, and the word problems that commonly appear in exams.

二次方程是 IGCSE 数学中考查频率最高的内容之一。无论 Paper 2 还是 Paper 4，无论是否允许使用计算器，你都会遇到它。本文系统讲解三种核心解法、判别式、函数图像、代数求交点问题，以及考试中常见的应用题型。

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

A quadratic equation is any equation that can be written in the standard form:

二次方程指任何可以写成标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants, and x is the variable. The highest power of x is 2, which is why the graph of a quadratic is a curve called a parabola.

其中 a、b、c 为常数，x 为变量。x 的最高次数是 2，因此二次函数的图像是一条称为抛物线的曲线。

In IGCSE, you are expected to solve quadratics where a = 1 (simple cases) and where a > 1 (harder cases). You must also understand that a quadratic equation has at most two real roots. Questions may ask you to choose between factorisation, the formula, or completing the square — so you need fluency in all three.

在 IGCSE 中，你需要会解 a = 1 的简单二次方程，也要会解 a > 1 的较难形式。同时必须理解，二次方程最多有两个实根。题目可能要求你在因式分解、求根公式和配方法中选择合适的方法——因此三种方法都要熟练。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has integer roots. The idea is simple: if two numbers multiply to give zero, then one of them must be zero.

当二次方程有整数根时，因式分解是最快捷的方法。核心思想很简单：如果两个数相乘等于零，那么其中至少一个必须为零。

For example, to solve x² − 5x + 6 = 0, factorise as (x − 2)(x − 3) = 0. Then x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

例如，解 x² − 5x + 6 = 0，分解为 (x − 2)(x − 3) = 0，则 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

If (x − p)(x − q) = 0, then x = p or x = q.

For quadratics with a > 1, you need to find factors of ac that add to b. For example, 2x² + 7x + 3 = 0: ac = 6, and 6 + 1 = 7, so split 7x as 6x + x and factor by grouping.

对于 a > 1 的二次方程，你需要找到 ac 的因子，使其相加等于 b。例如 2x² + 7x + 3 = 0：ac = 6，且 6 + 1 = 7，所以把 7x 拆为 6x + x，再用分组分解。

Common factorisation patterns you should memorise:

你需要熟记的常见分解模式：

• Difference of two squares: x² − a² = (x − a)(x + a) | 平方差：x² − a² = (x − a)(x + a)
• Perfect square: x² + 2ax + a² = (x + a)² | 完全平方：x² + 2ax + a² = (x + a)²
• Perfect square: x² − 2ax + a² = (x − a)² | 完全平方：x² − 2ax + a² = (x − a)²

Always rearrange the equation into the form ax² + bx + c = 0 before factorising. If a common factor exists, take it out first.

因式分解前，务必把方程整理为 ax² + bx + c = 0 的形式。若有公因子，先提取出来。

3. The Quadratic Formula | 求根公式法

When factorisation is difficult or impossible, use the quadratic formula. You are given this formula in the IGCSE formula sheet, but you must know how to substitute correctly and simplify the result.

当因式分解困难或不可行时，使用求根公式。IGCSE 公式表会给出该公式，但你必须会正确代入并化简结果。

x = (−b ± √(b² − 4ac)) / 2a

Let us solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5. Substitute into the formula:

我们来解 2x² + 3x − 5 = 0。这里 a = 2，b = 3，c = −5。代入

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