📚 Mastering Quadratic Equations for Edexcel IGCSE Mathematics | 精通Edexcel IGCSE数学二次方程
Quadratic equations are a fundamental topic in the Edexcel IGCSE Mathematics syllabus. Mastery of solving quadratics, understanding their graphs, and applying them to real-world problems is essential for achieving high grades. In this comprehensive revision guide, we will explore all key aspects, from factorisation to the quadratic formula, discriminant analysis, and tackling exam-style questions.
二次方程是Edexcel IGCSE数学教学大纲中的基础主题。掌握各种求解方法、理解其图形并能将知识应用于实际问题，对于取得高分至关重要。在本综合复习指南中，我们将深入探讨从因式分解到求根公式、判别式分析以及解答考试类题目的所有关键内容。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation in one variable is an equation that can be written in the standard form ax² + bx + c = 0 ant term. For example, 2x² + 3x - 5 = 0 is a quadratic equation with a = 2, b = 3, c = -5.
单变量二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。ax² 称为二次项，bx 为一次项，c 为常数项。例如，2x² + 3x - 5 = 0 是一个二次方程，其中 a = 2, b = 3, c = -5。
omes a linear equation. IGCSE questions often require you to rearrange an equation into this standard form before solving.
方程中变量的最高次幂必须为2。如果 a = 0，则方程退化为一次方程。IGCSE题目常要求先将方程整理为标准形式再进行求解。
the standard form helps you correctly identify the coefficients a, b and c for later methods.
再例如：4x² = 9x - 2 可整理为 4x² - 9x + 2 = 0。识别标准形式有助于你准确找出系数 a、b 和 c，以便后续使用其他方法。
2. Solving by Factorisation | 因式分解法求解
q = 0, then p = 0 or q = 0. Always ensure the equation is set to zero before factorising.
当二次表达式可以因式分解为两个一次因式的乘积时，因式分解是最快的方法。核心原理是零因子定律：如果 p × q = 0，则 p = 0 或 q = 0。因式分解前务必先将方程设为零。
(x - 2)(x - 3) = 0. By the null factor law, x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
示例：解方程 x² - 5x + 6 = 0。因式分解得 (x - 2)(x - 3) = 0。由零因子定律得 x - 2 = 0 或 x - 3 = 0，故 x = 2 或 x = 3。
+ x + 3 = 0, factor by grouping: 2x(x+3) + 1(x+3) = (2x+1)(x+3)=0, so x = -1/2 or x = -3.
如果首项系数 a 不为1，可使用十字相乘法（分裂中项）。对于 2x² + 7x + 3 = 0，找出两个数积为2×3=6且和为7，即6和1。改写为 2x² + 6x + x + 3 = 0，分组分解：2x(x+3) + 1(x+3) = (2x+1)(x+3)=0，得 x = -1/2 或 x = -3。
3. Solving by Completing the Square | 配方法求解
from which the solutions are easily found. This method is useful when the quadratic does not factorise easily and is also the basis for deriving the quadratic formula.
配方法将 ax² + bx + c = 0 转化为 a(x + p)² + q = 0 的形式，由此容易求出解。当二次方程不易因式分解时此方法很有用，也是推导求根公式的基础。
ent of x: (6/2)² = 9 to both sides: x² + 6x + 9 = 8. Write as (x + 3)² = 8. Then x + 3 = ±√8, so x = -3 ± 2√2.
对于 x² + 6x + 1 = 0，移常数项：x² + 6x = -1。两边加上一次项系数一半的平方 (6/2)² = 9：x² + 6x + 9 = 8。写为 (x + 3)² = 8。于是 x + 3 = ±√8，解得 x = -3 ± 2√2。
[(x - 2)² - 4] + 5 = 0 → 2(x - 2)² - 8 + 5 = 0 → 2(x - 2)² = 3, so (x - 2)² = 3/2, x = 2 ± √3/2.
当 a ≠ 1 时，先提取系数 a。对于 2x² - 8x + 5 = 0，可写成 2(x² - 4x) + 5 = 0。在括号内配方：x² - 4x + 4 - 4 得 2[(x - 2)² - 4] + 5 = 0 → 2(x - 2)² - 8 + 5 = 0 → 2(x - 2)² = 3，故 (x - 2)² = 3/2，x = 2 ± √3/2。
4. The Quadratic Formula | 二次方程求根公式
+ bx + c = 0. It is derived by completing the square on the general form. The formula is:
求根公式给出任意二次方程 ax² + bx + c = 0 的解，它由一般形式的配方法推导而得。公式如下：
x = (-b ± √b² - 4ac) / 2a
generally two solutions, one using the plus sign and one using the minus sign. To use the formula, identify a, b and c from the standard form, then substitute carefully.
符号 ± 表示通常有两个解，一个取加号，一个取减号。使用公式时，先确定标准形式中的 a、b 和 c，然后仔细代入。
Example: Solve 2x² - 4x - 3 = 0. Here a=2, b=-4, c=-3. Substitute: x = ( -(-4
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