📚 IGCSE Maths: Mastering Quadratic Equations | IGCSE 数学：彻底掌握二次方程
Quadratic equations are a core part of the IGCSE Mathematics syllabus. They appear in algebra, graph sketching, and modelled problems involving area or motion. Building a reliable method for factorising, completing the square, and using the quadratic formula will help you solve almost any quadratic question efficiently.
二次方程是 IGCSE 数学大纲的核心内容。它出现在代数、函数图像以及面积或运动建模问题中。掌握因式分解、配方法和求根公式等可靠方法，能帮助你高效解决几乎任何二次方程问题。
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 variable is 2. Its general form is shown below, where a, b, and c are constants and a ≠ 0.
二次方程是变量最高次数为 2 的多项式方程。其一般形式如下所示，其中 a、b、c 为常数且 a ≠ 0。
ax² + bx + c = 0
inear, not quadratic. For example, x² + 5x + 6 = 0 is quadratic, but 3x + 2 = 0 is linear.
条件 a ≠ 0 至关重要：如果 a = 0，该方程就变为一次方程，而不是二次方程。例如，x² + 5x + 6 = 0 是二次方程，而 3x + 2 = 0 是一次方程。
2. Standard Form and Coefficients | 标准形式与系数
Before solving, always rewrite the equation so that one side equals 0. Group any like terms and arrange the powers of x in descending order.
在求解之前，务必将方程整理为一边等于 0。合并同类项，并按 x 的幂次从高到低排列。
For example, 2x² - 8 = 3x can be rearranged into standard form as follows:
例如，2x² - 8 = 3x 可以整理为标准形式如下：
2x² - 3x - 8 = 0
Here the coefficient a is 2, the coefficient b is -3, and the constant term c is -8. Identifying these values correctly is the first step in both the quadratic formula and the discriminant.
这里系数 a 是 2，系数 b 是 -3，常数项 c 是 -8。正确识别这些值是使用求根公式和判别式的第一步。
3. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when a quadratic has rational roots. The procedure is: first write the equation in standard form, then factorise the left-hand side, and finally set each factor equal to zero.
当二次方程有有理根时，因式分解通常是最快的方法。步骤是：先将方程写成标准形式，然后对方程左边进行因式分解，最后令每个因式等于零。
ider x² + 5x + 6 = 0. The left side factorises as (x + 2)(x + 3), so the equation becomes:
以 x² + 5x + 6 = 0 为例。左边可分解为 (x + 2)(x + 3)，因此方程变为：
(x + 2)(x + 3) = 0
property, either x + 2 = 0 or x + 3 = 0. This gives x = -2 or x = -3 as the two solutions.
根据零乘积性质，要么 x + 2 = 0，要么 x + 3 = 0。由此得到两个解 x = -2 或 x = -3。
example, 2x² + 7x + 3 = 0 factorises into (2x + 1)(x + 3) = 0, giving x = -1/2 or x = -3.
当首项系数不为 1 时要小心。例如，2x² + 7x + 3 = 0 可分解为 (2x + 1)(x + 3) = 0，得到 x = -1/2 或 x = -3。
4. Solving by Completing the Square | 配方法
Completing the square is a powerful method that works for any quadratic equation. It is especially useful when the roots are irrational or when you need to find the turning point of a graph.
配方法对任何二次方程都有效，是一种强大的方法。当根为无理数或需要求图像的顶点时，配方法尤其有用。
To complete the square for x² + bx, add and subtract (b/2)². For example, for x² - 6x +
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