📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 求解二次方程：因式分解、配方法与求根公式
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graphs and real-world problems. Many candidates can expand brackets confidently, but solving quadratics requires a deeper understanding of factorisation, the zero product property, and the quadratic formula. In this article, we bring together the key methods you need to solve any quadratic equation accurately and efficiently.
二次方程贯穿 IGCSE 数学课程的始终，从代数运算到函数图像和实际应用题都会出现。许多考生能够熟练展开括号，但求解二次方程还需要深入理解因式分解、零积性质以及求根公式。本文将系统梳理求解二次方程的关键方法，帮助你准确、高效地解决各类二次方程。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the unknown is 2. The equation can be written in the general form ax² + bx + c = 0, where x is the variable, and a, b and c are constants with a ≠ 0.
二次方程是未知数的最高次数为 2 的多项式方程。它通常可以写成一般形式 ax² + bx + c = 0，其中 x 是未知数，a、b、c 是常数，且 a ≠ 0。
er than quadratic. For example, 3x² − 5x + 2 = 0 is quadratic, while 5x + 2 = 0 is linear.
如果 a 等于 0，x² 项就会消失，方程就变成一次方程而不是二次方程。例如，3x² − 5x + 2 = 0 是二次方程，而 5x + 2 = 0 是一次方程。
2. Standard Form and Coefficients | 标准形式与系数
Before attempting to solve a quadratic equation, you should rearrange it into standard form. This means collecting all terms on one side so that the other side equals zero, and writing the terms in descending powers of x: ax² + bx + c = 0.
在尝试解二次方程之前，应先把方程整理成标准形式。也就是说，把所有项移到等式一边，使另一边等于 0，并按 x 的降幂排列为 ax² + bx + c = 0。
Identify the coefficients a, b and c carefully, including their signs. For example, in 2x² − 7x + 3 = 0, we have a = 2, b = −7 and c = 3.
要仔细确定系数 a、b 和 c，包括它们的符号。例如，在 2x² − 7x + 3 = 0 中，a = 2，b = −7，c = 3。
ax² + bx + c = 0, a ≠ 0
3. Solving by Factorising | 因式分解法
When the quadratic expression on the left-hand side can be written as a product of two linear factors, factorising is usually the fastest method. You need two numbers that multiply to give ac and add to give b, then split the middle term and factor by grouping.
当左边的二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。你需要找到两个数，它们的乘积等于 ac，和等于 b，然后拆分中间项并进行分组分解。
For example, solve x² − 5x + 6 = 0 −5, c = 6. We need two numbers multiplying to 6 and adding to −5: they are −2 and −3. So:
例如，解 x² − 5x + 6 = 0。这里 a = 1，b = −5，c = 6。我们需要两个数，乘积为 6，和为 −5：这两个数是 −2 和 −3。因此：
x² − 5x + 6 = (x − 2)(x − 3) = 0
This gives x = 2 or x = 3. Always check by substituting back into the original equation.
由此得到 x = 2 或 x = 3。一定要把解代回原方程进行检验。
4. The Zero Product Property | 零积性质
The reason factorising works is the zero product property: if the product of two or more factors is zero, then at least one of the factors must be zero. In symbols, if pq = 0, then p = 0 or q = 0 (or both).
因式分解法之所以有效，是因为零积性质：如果两个或多个因式的乘积为零，那么至少有一个因式必须为零。用符号表示，如果 pq = 0，那么 p = 0 或 q = 0（或两者都为零）。
et equal to zero separately. This converts one quadratic equation into two simple linear equations.
当你把二次方程因式分解为 (x − p)(x − q) = 0
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