📚 Quadratic Equations and Functions | 二次方程与函数
Quadratic equations are central to the IGCSE Mathematics syllabus. You will need to solve them by factorising, using the quadratic formula, and completing the square. You will also need to understand their graphs, use the discriminant, and apply quadratics to real-life problems.
二次方程是 IGCSE 数学考纲的核心内容。你需要掌握因式分解法、求根公式法、配方法解二次方程，还需要理解二次函数的图像、运用判别式，并将二次方程应用到实际问题中。
This revision guide covers the key skills and common pitfalls in a clear, step-by-step way. Each section pairs an English explanation with a Chinese explanation so you can study in both languages and check your understanding.
本复习指南以清晰、逐步的方式涵盖关键技能和常见错误。每一节都同时提供英文和中文解释，方便你进行双语学习并检验自己的理解。
1. What is a quadratic equation? | 什么是二次方程？
are constants and a ≠ 0. If a were zero, the equation would become linear, not quadratic.
二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。如果 a 为零，方程就变成一次方程，而不是二次方程。
The highest power of x is 2, so a quadratic equation has at most two solutions, often called roots. These roots may be real and distinct, real and equal, or not real at all.
x 的最高次数是 2，因此二次方程最多有两个解，通常称为根。这些根可能是两个不同实数、两个相等实数，或者完全没有实数解。
− 7 = 0 and x² = 9. The last one can be rearranged to x² − 9 = 0, so it is also quadratic.
例如 x² − 5x + 6 = 0、2x² + 3x − 7 = 0 和 x² = 9。最后一个可以整理为 x² − 9 = 0，因此它也是二次方程。
2. Solving by factorising | 因式分解法
l to zero. For example, x² − 5x + 6 = 0 becomes (x − 2)(x − 3) = 0, giving x = 2 or x = 3.
当二次式可以写成两个一次括号的乘积时，我们可以令每个括号等于零。例如，x² − 5x + 6 = 0 可化为 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。
Always check by expanding the brackets to make sure the middle term is correct. For x² + bx + c, look for two numbers that multiply to give c and add to give b.
一定要通过展开括号来检验中间项是否正确。对于 x² + bx + c，寻找两个数，它们相乘得到 c，相加得到 b。
o 6 and add to 7: 6 and 1. Then 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
如果 x² 的系数不是 1，例如 2x² + 7x + 3，先将 a 和 c 相乘，然后拆中间项，再分组因式分解。这里 2 × 3 = 6，所以需要两个数相乘得 6 且相加得 7：即 6 和 1。于是 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
- Factorise fully: (2x + 1)(x + 3) = 0 | 完全因式分解：(2x + 1)(x + 3) = 0
- Set each factor to zero: 2x + 1 = 0 or x + 3 = 0 | 令每个因式为零：2x + 1 = 0 或 x + 3 = 0
- Solve: x = −1/2 or x = −3 | 解得：x = −1/2 或 x = −3
3. The quadratic formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is given by:
求根公式可以解任何二次方程 ax² + bx + c = 0。公式如下：
x = (−b ± √b² − 4ac) ÷ 2a
refully, including negative signs. For x² − 4x − 5 = 0, we have a = 1, b = −4, c = −5, so:
代入 a、b、c 的值时要格外小心，包括负号。对于 x² − 4x − 5 = 0，有 a = 1、b = −4、c = −5，所以：
x = (4 ± √16 + 20) ÷ 2 = (4 ± 6) ÷ 2
formula is especially useful when factorising is difficult, such as when the roots are decimals or surds.
由此得到 x = 5 或 x = −1。当因式分解困难时，例如根是小数或无理数时，求根公式特别有用。
- Write the equation in standard form. | 将方程写成标准形式。
- Identify a, b and c. | 确定 a、b、c。
- Substitute into the formula and simplify. | 代入公式并化简。
- Give both roots and check if one must be rejected in context. | 给出两个根，并检查在具体情境中是否需要舍去一个。
4. Completing the square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is very useful for finding the turning point of a quadratic graph and for solving equations that do not factorise neatly.
配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式。这对于求二次函数图像的顶点以及解不能整齐因式分解的方程非常有用。
For x² + 6x +
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