📚 Mastering Quadratics: Solving Quadratic Equations | 攻克二次方程全指南
Quadratic equations appear in almost every IGCSE Mathematics paper, either as a direct solving question or as a hidden step inside a longer problem. This revision guide covers the standard form, factorisation, completing the square, the quadratic formula and the discriminant, together with graphs, word problems and the mistakes that lose marks in the exam.
二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是直接求解，还是作为长题中的一个隐藏步骤。本复习指南涵盖标准形式、因式分解、配方法、求根公式与判别式，并讲解函数图象、应用题以及考试中常见的失分错误。
1. The Standard Form | 标准形式
A quadratic equation in one unknown is any equation that can be rearranged into the standard form shown below, where a, b and c are real numbers and a ≠ 0.
ax² + bx + c = 0
ation becomes linear, so the condition a ≠ 0 is essential. Always rearrange the equation so that the right-hand side is zero before attempting any solving method.
其中 a 是二次项 x² 的系数，b 是一次项 x 的系数，c 是常数项。若 a = 0，x² 项消失，方程退化为一次方程，因此条件 a ≠ 0 至关重要。在使用任何解法之前，务必先整理方程，使等号右边为 0。
uation a = 2, b = −3 and c = 5. Watch the signs carefully: the coefficient b is −3, not 3.
例如，2x² + 5 = 3x 必须改写为 2x² − 3x + 5 = 0。此时 a = 2，b = −3，c = 5。请注意符号：系数 b 是 −3，而不是 3。
2. Solving by Factorisation | 因式分解法
Factorisation is the fastest method when the left-hand side factorises neatly into two brackets. After writing the equation in standard form, look for two numbers whose product equals ac and whose sum equals b. Then split the middle term, factor by grouping, and apply the zero product property.
当等号左边可以顺利分解时，因式分解是最快的方法。将方程整理为标准形式后，寻找两个数，使它们的乘积等于 ac，和等于 b。然后拆分中间项、分组分解，并利用零积性质。
acket equal to zero and solve the resulting linear equations. Every step should be written down to avoid sign errors.
零积性质指出：若 p × q = 0，则 p = 0 或 q = 0。令每个括号等于零，再解所得的一次方程即可。每一步都要写清楚，以避免符号错误。
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3
m gives 2x² + 6x − x − 3 = 0, then grouping gives (2x − 1)(x + 3) = 0, so x = ½ or x = −3.
再看一例：2x² + 5x − 3 = 0。需要找两个数，乘积为 2 × (−3) = −6，和为 5；这两个数是 6 和 −1。拆分中间项得 2x² + 6x − x − 3 = 0，分组得 (2x − 1)(x + 3) = 0，所以 x = ½ 或 x = −3。
3. Difference of Two Squares and Special Patterns | 平方差与特殊模式
rks for numbers, variables and even more complex terms such as 4x² − 9 = (2x + 3)(2x − 3).
一种特殊的分解模式能在考试中节省宝贵时间：平方差公式。任何形如 a² − b² 的式子都可以分解为 (a + b)(a − b)。这个公式适用于数字、变量，甚至更复杂的项，例如 4x² − 9 = (2x + 3)(2x − 3)。
x² − 25 = 0 → (x + 5)(x − 5) = 0 → x = 5 or x = −5
and x² − 2px + p² = (x − p)². Recognising these patterns allows you to solve in one line.
注意，当常数项是负的完全平方数时，两根互为相反数。同时要记住完全平方模式：x² + 2px + p² = (x + p)²，x² − 2px + p² = (x − p)²。识别这些模式可以让你一步写出答案。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form (x + p)² + q. The key idea is that x² + bx can be turned into a perfect square by adding and subtracting (b ÷ 2)². This method is essential when a question asks for the turning point of a graph or when the roots involve surds.
配方法将二次式改写为 (x + p)² + q 的形式。核心思路是：通过加上并减去 (b ÷ 2)²，将 x² + bx 化为完全平方。当题目要求图象的顶点坐标，或根含有根号时，这种方法必不可少。
- Half the coefficient of x: b ÷ 2.
- Square the result: (b ÷ 2)².
- Add and subtract this value inside the expression.
- Factor the perfect square and simplify the constants.
- Solve by isolating the bracket and taking square roots, remembering the ± sign.
- 将 x 的系数除以 2：b ÷ 2。
- 将结果平方：(b ÷ 2)²。
- 在表达式中加上并减去这个值。
- 分解完全平方并化简常数项。
- 隔离平方项后开平方求解，不要忘记 ± 号。
x² + 6x + 2 = 0 → (x + 3)² − 9 + 2 = 0 → (x + 3)² = 7 → x = −3 ± √7
re completing the square. This keeps the arithmetic manageable and prevents common errors.
如果 x² 的系数不是 1，先将方程两边同时除以该系数。例如 2x² + 8x + 3 = 0 应先化为 x² + 4x + 1.5 = 0，再进行配方。这样可以简化计算并避免常见错误。
5. The Quadratic Formula | 求根公式
ions are always given by the formula below. Substitute a, b and c carefully, especially the signs.
求根公式适用于一切二次方程，包括
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