📚 Solving Quadratic Equations for IGCSE Edexcel Maths | IGCSE Edexcel 数学：解一元二次方程
Quadratic equations are a fundamental part of the IGCSE Edexcel Maths syllabus. They appear in algebraic manipulation, graph sketching, and real-world problem solving. Mastering the techniques to solve them—factorising, completing the square, and the quadratic formula—will strengthen your overall mathematical skills and prepare you for higher-level studies.
二次方程是 IGCSE Edexcel 数学大纲中的核心内容，涉及代数运算、函数图像描绘和实际应用问题的解决。熟练掌握因式分解法、配方法和求根公式这三种解法，不仅能提升你的综合数学能力，也为后续学习打下坚实基础。
1. What is a Quadratic Equation? | 什么是二次方程？
Quadratic equations often have two solutions, also called roots, which can be real or complex numbers, but at IGCSE level we focus on real roots.
二次方程是一个最高次数为 2 的多项式方程，通常以 x 为变量，标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。如果 a = 0，则方程退化为一元一次方程。二次方程通常有两个解，也称根，它们可能是实数或复数，但在 IGCSE 阶段我们主要关注实数根。
ax² + bx + c = 0
The solutions (roots) satisfy the equation, meaning when substituted back, the left-hand side equals zero. Recognising the standard form helps you decide which solving method to apply.
二次方程的解（根）能使方程成立，即代入后左边为零。识别标准形式有助于选择适当的解法。
2. Solving by Factorising: The Basics | 因式分解法基础
x² + 7x + 10 = 0 can be factorised as (x + 5)(x + 2) = 0, giving roots x = -5 and x = -2.
因式分解法是将 ax² + bx + c 写成 (px + q)(rx + s) 的形式。最简单的情况是 a = 1。我们需要找到两个数，它们的乘积等于常数项 c，其和等于一次项系数 b。例如 x² + 7x + 10 = 0 可以分解为 (x + 5)(x + 2) = 0，从而得到根 x = -5 和 x = -2。
iplying to −18 and adding to −3: −6 and +3. So (x − 6)(x + 3) = 0, roots x = 6 and x = −3.
务必先把方程化为等于零的形式。如 x² − 3x − 18 = 0，找到乘积为 −18，和为 −3 的两个数：−6 和 +3。于是 (x − 6)(x + 3) = 0，根为 x = 6 和 x = −3。
3. Factorising When a > 1 | a > 1 时的因式分解
+ 3) + 1(x + 3) = (2x + 1)(x + 3). Setting each factor to zero gives x = −1/2 and x = −3.
当二次项系数 a 不等于 1 时，分解需要更多步骤。以 2x² + 7x + 3 为例，先计算 a 与 c 的乘积 (2 × 3 = 6)，然后找两个数，乘积为 6，和为 7：即 6 和 1。将一次项拆成 6x + x，原式变为 2x² + 6x + x + 3。通过分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。令各因式为零，得 x = −1/2 和 x = −3。
Another approach is trial and error, but the splitting method is systematic. Practise recognising pairs quickly to save time.
另一种方法是试凑法，但拆项法更加系统化。通过练习快速找出数对，可以节省考试时间。
4. Special Cases: Difference of Two Squares | 特殊情况：平方差公式
A quadratic of the form x
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