📚 Quadratic Equations: Factorising and the Quadratic Formula | 二次方程：因式分解与求根公式
Quadratic equations appear throughout IGCSE Mathematics, from algebra and graphs to real-life modelling. This revision guide explains how to solve them by factorising, by the quadratic formula, and by completing the square, with worked examples and common pitfalls.
二次方程贯穿 IGCSE 数学，从代数、图像到现实建模都会出现。本复习指南讲解如何用因式分解法、二次公式法和配方法解二次方程，并提供例题和常见错误提醒。
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 ax² + bx + c = 0 ould become linear, not quadratic. Examples include x² − 7x + 10 = 0 and 2x² + 3x − 5 = 0.
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 是常数且 a ≠ 0。如果 a 为零，方程就变成一次方程，而不是二次方程。例如 x² − 7x + 10 = 0 和 2x² + 3x − 5 = 0。
The terms have special names: ax² is the quadratic term, bx is the linear term, and c is the constant term. IGCSE questions often ask you to rearrange an equation into this form before solving.
这些项有专门名称：ax² 是二次项，bx 是一次项，c 是常数项。IGCSE 题目常常要求先把方程整理成这种形式再求解。
2. The Zero Product Property | 零积性质
The key idea behind factorising is the zero product property: if two 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).
因式分解法的关键思想是零积性质：如果两个因式的乘积为零，那么至少有一个因式必须为零。用符号表示，如果 p × q = 0，那么 p = 0 或 q = 0（或两者都为零）。
This property is only valid when the product equals zero, so students should always set one side of the equation to zero before factorising.
这个性质只在乘积等于零时成立，所以学生一定要先把方程的一边化为零，再进行因式分解。
3. Solving by Factorising: Step by Step | 因式分解法步骤
nd side into two brackets. Third, set each bracket equal to zero. Finally, solve the two resulting linear equations.
用因式分解法解二次方程时，可以按照以下步骤进行。第一，把方程写成 ax² + bx + c = 0 的形式。第二，将左边分解成两个括号相乘。第三，令每个括号等于零。最后，解得到的两个一次方程。
For example, solve x² − 5x + 6 = 0. The left-hand side factorises as (x − 2)(x − 3) = 0 h factor to zero gives x − 2 = 0, so x = 2, and x − 3 = 0, so x = 3. The two solutions are x = 2 and x = 3.
例如，解 x² − 5x + 6 = 0。左边可以分解为 (x − 2)(x − 3) = 0。令每个因式为零，得到 x − 2 = 0，所以 x = 2；x − 3 = 0，所以 x = 3。两个解分别是 x = 2 和 x = 3。
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2, x = 3
4. Factorising When a > 1 | 二次项系数大于 1 的因式分解
When the coefficient of x² is not 1, factorising is slightly harder. You need to find two numbers that multiply to give ac and add to give b, then split the middle term or try bracket combinations systematically.
当 x² 的系数不是 1 时，因式分解会稍微困难一些。你需要找到两个数，使它们的乘积等于 ac，和等于 b，然后拆分中间项，或者系统地尝试括号组合。
For 2x² + 7x + 3 = 0 t 7x as 6x + x: 2x² + 6x + x + 3 = 0. Factorise in pairs: 2x(x + 3) + 1(x + 3) = 0, giving (2x + 1)(x + 3) = 0. Thus 2x + 1 = 0 gives x = −1/2, and x + 3 = 0 gives x = −3.
对于 2x² + 7x + 3 = 0，我们需要两个数，它们的乘积是 2 × 3 = 6，和是 7。这两个数是 6 和 1，所以把 7x 拆成 6x + x：2x² + 6x + x + 3 = 0。分组分解：2x(x + 3) + 1(x + 3) = 0，得到 (2x + 1)(x + 3) = 0。所以 2x + 1 = 0 得 x = −1/2，x + 3 = 0 得 x = −3。
5. Difference of Two Squares | 平方差
A special quadratic pattern is the difference of two squares: a² − b² = (a − b)(a + b). This appears frequently in IGCSE exams. For example, x² − 9 = 0 can be written as (x − 3)(x + 3) = 0, giving x = 3 or x = −3.
一种特殊的二次式规律是平方差：a² − b² = (a − b)(a + b)。这在 IGCSE 考试中经常出现。例如，x² − 9 = 0 可以写成 (x − 3)(x + 3) = 0，得到 x = 3 或 x = −3。
You can recognise this type when the quadratic has no bx term and the constant is negative. Always check whether both terms are perfect squares before applying the pattern.
当二次式没有 bx 项且常数项为负时，你可以识别出这种类型。在应用这个公式之前，一定要先检查两项是否都是完全平方数。
6. The Quadratic Formula | 二次公式
now the quadratic formula. For any quadratic equation ax² + bx + c = 0, the solutions are:
并不是每个二次式都能容易地因式分解，所以 IGCSE 学生还必须掌握二次公式。对于任意二次方程 ax² + bx + c = 0，解为：
x = (−b ± √b² − 4ac) / 2aPublished by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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