📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
A quadratic equation is one of the most important algebraic topics in IGCSE Mathematics. It appears in pure algebra questions, graph-sketching problems and real-life application tasks. This revision guide explains the core methods step by step, with bilingual notes to help you revise efficiently.
二次方程是 IGCSE 数学中最重要的代数主题之一。它出现在纯代数题、函数图像题和实际应用题中。本复习指南逐步讲解核心方法，并配有双语笔记，帮助你高效复习。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the unknown variable is 2. In IGCSE Mathematics, the unknown is usually written as x, but any letter can be used. Depending on the values of its coefficients, a quadratic equation can have two distinct real solutions, one repeated real solution, or no real solutions at all.
二次方程是未知数的最高次数为 2 的多项式方程。在 IGCSE 数学中，未知数通常写作 x，但也可以使用任何字母。根据方程系数的不同，二次方程可能有两个不相等的实根、一个重根，或者没有实根。
ax² + bx + c = 0, a ≠ 0
This standard form is the starting point for nearly every solution method you will meet in the course.
这个标准形式是你在课程中几乎所有解法方法的出发点。
2. Standard Form and Coefficients | 标准形式与系数
r rather than quadratic. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.
二次方程的一般标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不等于 0。条件 a ≠ 0 非常重要，因为如果 a = 0，方程就变成一次方程而不是二次方程。ax² 称为二次项，bx 称为一次项，c 称为常数项。
0, the missing linear term means b = 0, so the equation can be written as x² + 0x + 7 = 0.
正确识别系数是大多数解法的第一步。例如，在方程 3x² − 5x + 2 = 0 中，a = 3，b = −5，c = 2。在方程 x² + 7 = 0 中，一次项缺失，说明 b = 0，因此该方程可以写作 x² + 0x + 7 = 0。
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when a quadratic equation has simple integer roots. The expression ax² + bx + c is rewritten as the product of two linear factors, usually in the form (px + q)(rx + s). Once the equation is factorised, each factor can be set to zero to find the possible values of x.
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 写成两个一次因式的乘积，通常形式为 (px + q)(rx + s)。方程因式分解后，可以令每个因式等于零来求出 x 的可能值。
bers are −2 and −3, so the equation becomes (x − 2)(x − 3) = 0. This gives x = 2 or x = 3.
例如，求解 x² − 5x + 6 = 0 时，需要找两个数，使它们的积为 6，和为 −5。这两个数是 −2 和 −3，因此方程变为 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。
Always rearrange the equation into standard form before factorising. If the equation is not equal to zero, factorising alone will not give correct solutions.
因式分解前一定要将方程化为标准形式。如果方程右边不是零，仅靠因式分解无法得到正确的解。
4. Zero Product Property | 零积性质
imple logical rule is the reason why the factorising method works for quadratic equations.
零积性质指出，如果两个数的乘积为零，那么其中至少一个数必须为零。用符号表示就是：若 pq = 0，则 p = 0 或 q = 0。这一简单逻辑规则正是因式分解法能够求解二次方程的原因。
oduce two distinct answers, unlike a linear equation which normally has only one solution.
例如，若 (2x − 1)(x + 4) = 0，则 2x − 1 = 0 或 x + 4 = 0。解这两个一次方程得到 x = ½ 和 x = −4。注意，一个二次方程可以产生两个不同的答案，而一次方程通常只有一个解。
5. Completing the Square | 配方法
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