📚 Algebra 9: Quadratic Equations & Simultaneous Equations | 代数9：二次方程与联立方程组
Quadratic equations and simultaneous equations are central topics in the Edexcel IGCSE Mathematics syllabus. They appear frequently in both Foundation and Higher Tier papers, often combining algebraic manipulation with graphical interpretation and real-world problem solving. Mastering these topics is essential for achieving top scores in algebra.
二次方程与联立方程组是Edexcel IGCSE数学大纲中的核心内容，经常出现在基础卷和进阶卷中，通常涉及代数运算、图像解释和实际问题求解。掌握这些主题是在代数部分取得高分的关键。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2. The standard form is ax² + bx + c = 0 and c are constants and a ≠ 0. If a = 0, the equation becomes linear. Examples include x² – 5x + 6 = 0 and 3x² + 2x – 1 = 0.
二次方程是次数为2的多项式方程。标准式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。若 a = 0，方程退化为一次方程。例子有 x² – 5x + 6 = 0 和 3x² + 2x – 1 = 0。
y = ax² + bx + c is a parabola, and the roots correspond to the x-intercepts of the graph.
二次方程的解称为根。根可以是实数或复数；在 IGCSE 阶段我们只关注实数根。二次函数 y = ax² + bx + c 的图像是一条抛物线，方程的根即为图像与 x 轴交点的横坐标。
2. Solving by Factorisation | 因式分解法
so the factorised form is (x + 2)(x + 3) = 0. Setting each factor equal to zero gives x = –2 or x = –3.
当二次式可分解为两个一次因式的乘积时，因式分解法是最快捷的方法。例如解 x² + 5x + 6 = 0，找到两个数乘积为 6、和为 5，即 2 和 3，因此因式分解为 (x + 2)(x + 3) = 0。令每个因式为零得 x = –2 或 x = –3。
6x + x + 3 = 0. Factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. Hence, x = –1/2 or x = –3.
对于 a ≠ 1 的二次方程，如 2x² + 7x + 3 = 0，可用拆分中项法：找到两个数乘积为 ac（2×3=6）且和为 7，即 6 和 1。将 7x 改为 6x + x：2x² + 6x + x + 3 = 0。分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0，解得 x = –1/2 或 x = –3。
3. Completing the Square | 配方法
ider x² + 6x + 5 = 0. Half of 6 is 3, square it to get 9. Rewrite the equation as (x + 3)² – 9 + 5 = 0 → (x + 3)² – 4 = 0 → (x +
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