📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in many exam questions, both as standalone problems and as part of larger questions about graphs, areas, and coordinate geometry. In this article, we will review the standard form of a quadratic equation, the three main solution methods, the discriminant, and common exam pitfalls.
二次方程是 IGCSE 数学大纲中最重要的主题之一。它们出现在许多考试题目中，既作为独立问题，也作为图像、面积和坐标几何等更大问题的一部分。在本文中，我们将复习二次方程的标准形式、三种主要的解题方法、判别式以及常见的考试陷阱。
1. Understanding Quadratic Equations | 理解二次方程
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 value of a cannot be zero, because then the equation would become linear and lose its quadratic nature.
二次方程是任何可以写成标准形式ax² + bx + c = 0的方程，其中 a、b、c 是常数且 a ≠ 0。a 的值不能为零，否则方程将变成一次方程，从而失去二次特性。
The highest power of the variable is 2, which is why the equation is called “quadratic”. In IGCSE Mathematics, you will meet three main methods for solving quadratics: factorisation, the quadratic formula, and completing the square. Each method has its own strengths, and the table below summarises when to use each one.
变量的最高次数是 2，因此该方程被称为“二次”方程。在 IGCSE 数学中，你会遇到三种主要的求解二次方程的方法：因式分解法、求根公式法和配方法。每种方法各有优势，下表总结了各自的最佳使用时机。
| Method / 方法 | Best used when / 最佳使用时机 |
|---|---|
| Factorisation / 因式分解法 | The equation factorises easily with integers / 方程能用整数轻松分解时 |
| Quadratic formula / 求根公式法 | The equation does not factorise neatly / 方程不易整齐分解时 |
| Completing the square / 配方法 | You need the turning point or exact surd form / 需要顶点坐标或精确根式形式时 |
2. Solving by Factorisation (a = 1) | 因式分解法（a = 1）
When the coefficient of x² is 1, we look for two numbers whose product is c and whose sum is b. The equation then factors as (x + m)(x + n) = 0. This is the quickest method when the numbers are easy to find.
当 x² 的系数为 1 时，我们寻找两个数，其乘积等于 c，和等于 b。这样方程就可以分解为 (x + m)(x + n) = 0。当数字容易找到时，这是最快的方法。
Example: Solve x² − 5x + 6 = 0. We need two numbers with product 6 and sum −5. Those numbers are −2 and −3, because (−2) × (−3) = 6 and (−2) + (−3) = −5.
例：解 x² − 5x + 6 = 0。我们需要找两个数，乘积为 6，和为 −5。这两个数是 −2 和 −3，因为 (−2) × (−3) = 6 且 (−2) + (−3) = −5。
(x − 2)(x − 3) = 0 → x = 2 or x = 3
Use the zero product property: if the product of two factors is zero, then at least one factor must be zero. Hence x − 2 = 0 or x − 3 = 0. Always check your factors by expanding them back.
使用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式必须为零。因此 x − 2 = 0 或 x − 3 = 0。务必通过展开来检验你的因式是否正确。
3. Solving by Factorisation (a ≠ 1) | 因式分解法（a ≠ 1）
When a is not 1, for example 2x² + 7x + 3 = 0, we first multiply a and c. Here 2 × 3 = 6. We then find two numbers whose product is 6 and whose sum is 7; those numbers are 6 and 1.
当 a 不等于 1 时，例如 2x² + 7x + 3 = 0，我们先将 a 与 c 相乘。这里 2 × 3 = 6。然后找两个数，其乘积为 6，和为 7；这两个数是 6 和 1。
Rewrite the middle term using these two numbers: 2x² + 6x + x + 3 = 0. Then factor by grouping: 2x(x + 3) + 1(x + 3) = 0. Notice that (x + 3) is a common factor, so we get (2x + 1)(x + 3) = 0.
用这两个数重写中间项：2x² + 6x + x + 3 = 0。然后分组因式分解：2x(x + 3) + 1(x + 3) = 0。注意 (x + 3) 是公因式，因此得到 (2x + 1)(x + 3) = 0。
2x + 1 = 0 or x + 3 = 0 → x = −½ or x = −3
This method is often called the “ac method” because you multiply a by c at the start. It works for any quadratic that factorises over the integers.
这种方法通常称为“ac 法”，因为一开始需要将 a 乘以 c。它适用于所有可以在整数范围内因式分解的二次方程。
4. The Quadratic Formula | 求根公式法
Some quadratic equations cannot be factorised easily, especially when the roots involve surds or decimals. The quadratic formula works for every quadratic equation and is given on the IGCSE formula sheet:
有些二次方程不容易因式分解，尤其当根涉及根式或小数时。求根公式适用于每一个二次方程，并且在 IGCSE 公式表中给出：
x = (−b ± √(b² − 4ac)) / (2a)
Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute carefully, keeping the negative signs:
例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入时要小心，保留负号：
x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4
Simplify the surd: √40 = √(4 × 10) = 2√10. Therefore x = (4 ± 2√10) / 4 = 1 ± √10/2. Give your answer in exact surd form, or as a decimal rounded to 3 significant figures if the question asks for it.
化简根式：√40 = √(4 × 10) = 2√10。因此 x = (4 ± 2√10) / 4 = 1 ± √10/2。答案用精确根式形式给出，如果题目要求，也可以化为保留 3 位有效数字的小数。
5. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This form is especially useful for finding the turning point of a parabola and for solving equations without using the formula.
配方法将 ax² + bx + c 改写成 a(x + p)² + q 的形式。这种形式特别适用于求抛物线的顶点坐标，以及不用求根公式来解方程。
Step 1: Halve the coefficient of x. For x² +
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