📚 Quadratic Equations: Solving, Graphs and Applications | 二次方程：求解、图像与应用
Quadratic equations are a central topic in IGCSE Mathematics. They link algebra with graphing and often appear in multi-step problem solving. This guide covers the main methods, graph features and common exam pitfalls.
二次方程是 IGCSE 数学的核心主题。它将代数与图像联系起来，经常出现在多步骤解题中。本指南涵盖主要求解方法、图像特征和常见考试陷阱。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
c are constants and a ≠ 0. The condition a ≠ 0 is essential because the x² term must be present.
一元二次方程总是可以整理成标准形式 ax² + bx + c = 0，其中 a、b 和 c 为常数且 a ≠ 0。条件 a ≠ 0 非常重要，因为 x² 项必须存在。
e coefficient of x² is zero, the equation reduces to a linear equation such as bx + c = 0.
例如，2x² - 3x + 1 = 0 中 a = 2，b = -3，c = 1。如果 x² 的系数为零，方程就会退化为 bx + c = 0 这样的一次方程。
2. Solving by Factorisation | 因式分解法求解
Factorisation is usually the fastest method when the quadratic has simple integer roots. Write ax² + bx + c as a product of two brackets, then use the fact that if a product equals zero, at least one factor must be zero.
当二次方程具有简单的整数根时，因式分解通常是最快的方法。将 ax² + bx + c 写成两个括号的乘积，然后利用若乘积为零则至少有一个因式为零的性质。
the factorisation is (x - 2)(x - 3) = 0. Setting each factor to zero gives x = 2 or x = 3.
例如，求解 x² - 5x + 6 = 0 时，我们寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3，因此分解为 (x - 2)(x - 3) = 0。令每个因式为零得到 x = 2 或 x = 3。
Always expand your brackets as a check. A sign mistake is one of the most common errors in IGCSE papers.
务必展开括号进行检验。符号错误是 IGCSE 试卷中最常见的错误之一。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x - h)² + k, which is useful for finding the turning point and for solving when factorisation is difficult. The key identity for a monic quadratic is:
配方法将二次式改写为 a(x - h)² + k 的形式，这在寻找转折点以及因式分解困难时求解非常有用。对于首项系数为 1 的二次式，关键恒等式为：
x² + bx = (x + b/2)² - (b/2)²
e x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4. This shows the turning point is (-3, -4).
例如，对 x² + 6x + 5 配方。取 6 的一半，即 3，然后写出 x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4。这表明转折点为 (-3, -4)。
If a ≠ 1, factor a out of the x² and x terms first, then complete the square inside the bracket. Do not change the constant term without adjusting correctly.
如果 a ≠ 1，应先将 a 从 x² 项和 x 项中提出，然后在括号内完成配方。不要在不正确调整常数项的情况下改变它。
4. The Quadratic Formula | 二次公式
ot integers. It is derived by completing the square on the standard form ax² + bx + c = 0.
二次公式可以求解任何二次方程，即使根是无理数或不是整数。它是由标准形式 ax² + bx + c = 0 配方法推导出来的。
x = (-b ± √b² - 4ac) / (2a)
For example, to solve 2x² + 3x - 2 = 0, substitute a = 2, b = 3 and c = -2. Then x = (-3 ± √9 + 16) / 4 = (-3 ± 5) / 4, giving x = 1/2 or x = -2.
例如，求解 2x² + 3x - 2 = 0 时，代入 a = 2，b = 3，c = -2。则 x = (-3
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