📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解一元二次方程：因式分解、配方法与求根公式
Quadratic equations appear throughout the IGCSE syllabus, from algebraic manipulation to geometry and real-world modelling. This article explains the three core algebraic methods for solving quadratic equations: factorising, completing the square and using the quadratic formula. Each method is illustrated with worked examples and linked to the skills examiners expect.
一元二次方程贯穿 IGCSE 数学大纲，从代数变形到几何与实际建模都有出现。本文讲解解一元二次方程的三种核心代数方法：因式分解法、配方法和求根公式法，并结合例题说明考官要求的技能。
1. What is a Quadratic Equation? | 什么是二次方程？
s and a ≠ 0. The highest power of the unknown x is 2, which is why it is called quadratic.
一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。未知数 x 的最高次数是 2，因此称为二次方程。
The coefficient a is the coefficient of x², b is the coefficient of x, and c is the constant term. In IGCSE questions, you may need to rearrange an equation into this standard form before solving.
系数 a 是 x² 的系数，b 是 x 的系数，c 是常数项。在 IGCSE 题目中，你可能需要先把方程整理成标准形式再求解。
2. Standard Form and Key Terms | 标准形式与关键术语
are ordered by descending powers of x. For example, 3x + 2x² = 5 becomes 2x² + 3x − 5 = 0.
总是将方程整理为一边为零、各项按 x 的降幂排列。例如 3x + 2x² = 5 变为 2x² + 3x − 5 = 0。
Key terms include the leading coefficient, the constant term, and the roots or solutions of the equation. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots.
关键术语包括首项系数、常数项以及方程的根或解。一元二次方程可能有两个不同实根、一个重根或没有实根。
It is important to distinguish between a quadratic expression, such as ax² + bx + c, and a quadratic equation, where this expression is set equal to zero. Only an equation can be solved for x.
区分二次式与二次方程很重要：二次式如 ax² + bx + c 本身只是一个表达式，而二次方程是将这个表达式设为零。只有方程才能解出 x。
3. Solving by Factorising: The Basic Idea | 因式分解法的基本思想
x + q)(rx + s) = 0, then at least one factor must equal zero, so px + q = 0 or rx + s = 0.
因式分解法将二次式写成两个一次因式的乘积。如果 (px + q)(rx + s) = 0，那么至少有一个因式等于零，所以 px + q = 0 或 rx + s = 0。
This method works quickly when the roots are rational and the coefficients are small. You look for two numbers that multiply to give ac and add to give b, then split the middle term.
当根为有理数且系数较小时，这种方法最快。你寻找两个数，使它们的乘积为 ac、和为 b，然后拆分中间项。
eed two numbers whose product is c and whose sum is b. For example, to factorise x² − 5x + 6, the numbers −2 and −3 multiply to 6 and add to −5.
x² − 5x + 6 = (x − 2)(x − 3)
对于 a = 1 的简单二次式，你需要两个数，它们的乘积为 c、和为 b。例如，对 x² − 5x + 6 进行因式分解时，−2 和 −3 的乘积为 6、和为 −5。
x² − 5x + 6 = (x − 2)(x − 3)
4. Worked Example of Factorising | 因式分解法例题
+ 7x + 12 = 0. We need two numbers whose product is 12 and sum is 7; they are 3 and 4. So:
解方程 x² + 7x + 12 = 0。我们需要两个数，乘积为 12、和为 7；它们是 3 和 4。因此：
x² + 7x + 12 = (x + 3)(x + 4) = 0
x² + 7x + 12 = (x + 3)(x + 4) = 0
Setting each factor to zero gives x + 3 = 0 or x + 4 = 0, so x = −3 or x = −4.
令每个因式为零，得 x + 3 = 0 或 x + 4 = 0，所以 x = −3 或 x = −4。
o numbers with product 2 × (−3) = −6 and sum 5; these are 6 and −1. Split the middle term:
另一个首项系数不为 1 的例子：解 2x² + 5x − 3 = 0。找两个数，乘积为 2 × (−3) = −6、和为 5，这两个数是 6 和 −1。拆分中间项：
2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3) = 0
2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3) = 0
Thus 2x − 1 = 0 or x + 3 = 0, giving x = 1/2 or x = −3.
因此 2x − 1 = 0 或 x + 3 = 0，得 x = 1/2 或 x = −3。
5. When Factorising is Not Possible | 因式分解不适用时
s. For example, x² − 4x + 1 = 0 has no obvious pair of integers with product 1 and sum −4.
有些二次式不能分解为简单的整数因式。例如 x² − 4x + 1 = 0 没有明显的整数对满足乘积为 1、和为 −4。
In these cases, you can use completing the square or the quadratic formula. The method of completing the square also helps find the turning point of a parabola, which is useful in graph-sketching questions.
这时可以使用配方法或求根公式。配方法还能帮助求出抛物线的顶点，这对画图题很有用。
Do not waste time looking for factors that do not exist. If the discriminant b² − 4ac is not a perfect square, the roots are irrational and factorising over integers will not work.
不要浪费时间寻找不存在的因式。如果判别式 b² − 4ac 不是完全平方数，那么根是无理数，无法在整数范围内进行因式分解。
6. Completing the Square | 配方法
To complete the square for x² + bx, add and subtract (b/2)² to create a perfect square. In general:
对 x² + bx 配方时，通过加上并减去 (b/2)² 构造完全平方。一般地：
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