📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与二次公式
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching and real-life modelling. This revision article explains how to identify a quadratic equation, solve it by factorising, completing the square and using the quadratic formula, and how to interpret the discriminant. Worked examples and common errors are included to help you feel confident in the exam.
二次方程贯穿 IGCSE 数学考纲，从代数变形到图像绘制和实际建模均有涉及。本篇复习文章讲解如何识别二次方程，如何运用因式分解法、配方法和二次公式求解，以及如何解释判别式。文中配有例题与常见错误，帮助你在考试中更有把握。
1. What is a Quadratic Equation? | 什么是二次方程？
bx + c = 0, where a ≠ 0. If the x² term is missing, the equation is linear, not quadratic.
二次方程是未知数最高次数为 2 的方程。在 IGCSE 数学中，变量通常为 x，方程常写成 ax² + bx + c = 0 的形式，其中 a ≠ 0。如果没有 x² 项，就是一次方程，而不是二次方程。
is also quadratic, even though the x term is absent, because the highest power is still 2.
例如，2x² + 5x - 3 = 0 是二次方程，因为 2x² 项的次数为 2。x² - 9 = 0 也是二次方程，尽管没有 x 项，因为最高次数仍然是 2。
2. Standard Form and Coefficients | 标准形式与系数
. The numbers a, b and c are called coefficients: a is the coefficient of x², b is the coefficient of x, and c is the constant term. Identifying a, b and c correctly is essential for the quadratic formula and the discriminant.
使用大多数方法求解二次方程之前，应先将其整理成标准形式 ax² + bx + c = 0。数 a、b 和 c 称为系数：a 是 x² 的系数，b 是 x 的系数，c 是常数项。正确识别 a、b 和 c 对二次公式和判别式至关重要。
and c = 2. Be careful with signs: negative coefficients must be carried into the formula.
在方程 3x² - 7x + 2 = 0 中，a = 3，b = -7，c = 2。注意符号：负系数必须代入公式。
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets. For a simple monic quadratic like x² + bx + c, look for two numbers that multiply to c and add to b. For non-monic quadratics, use splitting the middle term or trial and error.
当二次式能写成两个一次括号的乘积时，因式分解通常是最快的方法。对于 x² + bx + c 这类首项系数为 1 的二次式，找两个数，使它们的乘积为 c，和为 b。对于首项系数不为 1 的二次式，可以用中间项拆分或试错法。
he numbers 2 and 3 multiply to 6 and add to 5, so the equation becomes (x + 2)(x + 3) = 0.
例题：解方程 x² + 5x + 6 = 0。2 和 3 的乘积为 6，和为 5，因此方程变形为 (x + 2)(x + 3) = 0。
nding: 2x·x = 2x², 2x·3 + 1·x = 6x + x = 7x, and 1·3 = 3, so the factorisation is correct.
对于 2x² + 7x + 3 = 0 这类首项系数不为 1 的二次式，可以写成 (2x + 1)(x + 3) = 0。展开检验：2x·x = 2x²，2x·3 + 1·x = 6x + x = 7x，1·3 = 3，因此因式分解正确。
4. The Zero Product Property | 零乘积性质
set each bracket equal to zero and solve: x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
因式分解后，关键思想是零乘积性质：如果两个或多个因式的乘积为零，那么至少有一个因式为零。因此，由 (x + 2)(x + 3) = 0，可令每个括号等于零并求解：x + 2 = 0 或 x + 3 = 0，得 x = -2 或 x = -3。
This property only works when one side of the equation is exactly zero. Do not set each factor equal to a non-zero number; the equation must be rearranged to equal zero first.
这个性质只在方程的一边恰好为零时成立。不要把每个因式等于一个非零数；必须先把方程整理成等于零的形式。
5. Completing the Square | 配方法
Completing the square rewrites the quadratic expression in the form a(x + p)² + q, which makes the turning point of the graph easy to find. To complete the square for x² + bx + c, take half of b, square it, and add and subtract that value inside the bracket.
配方法将二次式改写成 a(x + p)² + q 的形式，这样图像的顶点就很容易找到。对 x² + bx + c 进行配方时，取 b 的一半，平方，并在括号内加减这个值。
x² + 6x + 9 - 9 + 5 = (x + 3)² - 4. The equation x² + 6x + 5 = 0 becomes (x + 3)² - 4 = 0.
对于 x² + 6x + 5，6 的一半为 3，3² = 9。改写为 x² + 6x + 9 - 9 + 5 = (x + 3)²
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