📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程精讲
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, problem-solving, and even in geometry and trigonometry contexts. Understanding how to solve them by factorisation, completing the square, and the quadratic formula is essential for achieving a high grade. In this article, we will explore each method, practise with worked examples, and highlight common mistakes to avoid.
二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、应用题，甚至几何和三角学情境中。掌握因式分解法、配方法和求根公式法求解二次方程，对取得高分至关重要。本文将逐一讲解每种方法、通过例题练习，并指出常见错误以避免失分。
1. Quadratic Equations and Their Standard Form | 二次方程及其标准形式
uation becomes linear, not quadratic. The coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term.
二次方程是任何可以整理成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。a ≠ 0 这个条件很重要，因为如果 a = 0，方程就变成一次方程而不是二次方程。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。
ax² + bx + c = 0, a ≠ 0
dratic at first, but after expanding brackets or clearing fractions, they can be written in standard form.
二次方程的例子包括 x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 4x² - 9 = 0。有些方程一开始看起来不像二次方程，但经过展开括号或去分母后，就可以写成标准形式。
2. Solving by Factorisation | 因式分解法求解
we apply the zero product property: if a product of two factors equals zero, then at least one factor must be zero.
当二次表达式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。要用因式分解法解 ax² + bx + c = 0，我们需要找到两个数，它们的乘积等于 ac，和等于 b，或者直接分解表达式。得到 (px + q)(rx + s) = 0 后，应用零乘积性质：如果两个因式的乘积等于零，那么至少有一个因式必须为零。
(x + 3) = 0. Set each factor to zero: x + 2 = 0 or x + 3 = 0. Solutions: x = -2 or x = -3.
例题：解 x² + 5x + 6 = 0。因式分解：(x + 2)(x + 3) = 0。令每个因式为零：x + 2 = 0 或 x + 3 = 0。解为：x = -2 或 x = -3。
Always check your factors by expanding them back to the original expression. This helps you catch sign errors before moving on.
一定要通过将因式展开回原表达式来检查你的因式。这可以帮助你在继续之前发现符号错误。
3. Solving by Completing the Square | 配方法求解
Completing the square is a powerful method that works for any quadratic equation, even when factorisation is difficult or impossible. The idea is to rewrite ax² + bx + c in the form a(x + p)² + q. For a monic quadratic x² + bx + c, we take half the coefficient of x, square it, and add and subtract that value. This creates a perfect square trinomial.
配方法是一种强大的方法，适用于任何二次方程，即使因式分解困难或不可能时也可以使用。其思想是将 ax² + bx + c 改写成 a(x + p)² + q 的形式。对于首一二次式 x² + bx + c，我们取 x 项系数的一半，平方，然后加上并减去这个值。这样就构成了完全平方三项式。
x² + bx = (x + b/2)² - (b/2)²Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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