Solving Quadratic Equations: Factorisation, Formula, and Graphs | 一元二次方程求解：因式分解法、公式法与图像法

📚 Solving Quadratic Equations: Factorisation, Formula, and Graphs | 一元二次方程求解：因式分解法、公式法与图像法

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching. Understanding how to solve them efficiently is essential for success in both Core and Extended papers.

一元二次方程贯穿 IGCSE 数学大纲，从代数变形到函数图像都会涉及。掌握高效求解方法对通过核心卷和扩展卷都至关重要。

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

A quadratic equation is an equation of degree 2, meaning the highest power of the unknown is 2. In its standard form, it is written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The condition a ≠ 0 matters because if a were 0, the equation would become linear, not quadratic.

一元二次方程是次数为 2 的方程，即未知数的最高次数是 2。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。a ≠ 0 这一条件很重要，因为如果 a = 0，方程就变成了一元一次方程，而不是二次方程。

ax² + bx + c = 0

Common examples include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0, and x² − 9 = 0. Notice that the equation must be set equal to zero before most solution methods can be applied.

常见例子有 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 以及 x² − 9 = 0。注意，在应用大多数求解方法之前，必须先把方程整理为等于 0 的形式。

2. Standard Form and Key Vocabulary | 标准形式与关键术语

Standard form is ax² + bx + c = 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term. The coefficient a is the leading coefficient; b is the linear coefficient. Being able to identify a, b, and c quickly is the first step in any method, especially for the quadratic formula.

标准形式为 ax² + bx + c = 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。系数 a 是二次项系数，b 是一次项系数。快速识别 a、b、c 是使用任何方法的第一步，尤其是使用求根公式时。

For example, in 2x² − 7x + 3 = 0, we have a = 2, b = −7, and c = 3. In 4x² − 9 = 0, the linear coefficient b is 0, so a = 4, b = 0, and c = −9.

例如，在 2x² − 7x + 3 = 0 中，a = 2，b = −7，c = 3。在 4x² − 9 = 0 中，一次项系数 b = 0，因此 a = 4，b = 0，c = −9。

3. Zero Product Property: The Foundation of Factorisation | 零乘积性质：因式分解的基础

The key idea behind solving by factorisation is the zero product property: if two or more factors multiply to give zero, then at least one of the factors must be zero. In symbols, if p × q = 0, then p = 0 or q = 0 (or both). This allows us to break a quadratic product into two smaller linear equations.

因式分解法求解的核心思想是零乘积性质：如果两个或多个因式相乘得 0，那么其中至少有一个因式必须为 0。用符号表示，若 p × q = 0，则 p = 0 或 q = 0（或两者都为 0）。这使我们可以把二次方程的乘积拆成两个较简单的一次方程。

This property only works after the quadratic expression has been factorised completely and the equation equals zero. It is not valid if the product equals a non-zero number.

该性质只有在二次表达式被完全因式分解且方程等于 0 时才成立。如果乘积等于非零数，则不能直接使用。

4. Solving by Factorisation: Step-by-Step Method | 因式分解法：分步求解

To solve ax² + bx + c = 0 by factorisation, follow these steps: (1) write the equation in standard form with zero on one side; (2) factorise the quadratic expression into two brackets; (3) set each bracket equal to zero; (4) solve the resulting linear equations; (5) check your answers by substitution.

用因式分解法解 ax² + bx + c = 0 的步骤如下：(1) 将方程写成一边为 0 的标准形式；(2) 将二次表达式因式分解成两个括号的乘积；(3) 令每个括号分别等于 0；(4) 解所得的一元一次方程；(5) 将答案代回原方程检验。

For simple quadratics where a = 1, you need two numbers that multiply to c and add to b. For example, for x² − 5x + 6, the numbers −2 and −3 multiply to 6 and add to −5, so the factorisation is (x − 2)(x − 3).

对于 a = 1 的简单二次式，需要找到两个数，使它们相乘等于 c、相加等于 b。例如，对于 x² − 5x + 6，−2 和 −3 相乘得 6、相加得 −5，因此因式分解为 (x − 2)(x − 3)。

5. Worked Example: Simple Factorisation | 实例：简单因式分解

Solve x² − 5x + 6 = 0. First, factorise: (x − 2)(x − 3) = 0. By the zero product property, x − 2 = 0 or x − 3 = 0. Therefore x = 2 or x = 3. Substituting x = 2 gives 4 − 10 + 6 = 0, and x = 3 gives 9 − 15 + 6 = 0, so both solutions are correct.

解方程 x² − 5x + 6 = 0。首先因式分解：(x − 2)(x − 3) = 0。根据零乘积性质，x − 2 = 0 或 x − 3 = 0。因此 x = 2 或 x = 3。将 x = 2 代入得 4 − 10 + 6 = 0，将 x = 3 代入得 9 − 15 + 6 = 0，所以两个解都正确。

Notice that a quadratic equation can have up to two real solutions. In this case there are exactly two distinct solutions. This is the most common scenario in IGCSE questions.

注意，一元二次方程最多可以有两个实数解。这里正好有两个不同的解。这是 IGCSE 题目中最常见的情形。

6. Worked Example: a ≠ 1 and

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