📚 Mastering Quadratic Equations for IGCSE Maths | 掌握 IGCSE 数学二次方程
Quadratic equations run through the whole IGCSE Mathematics syllabus, from algebra and graphs to coordinate geometry and problem solving. This revision guide brings together the essential methods, the key facts and the common exam traps in one place, written in English and Chinese so you can build precise mathematical language in both.
二次方程贯穿 IGCSE 数学大纲，从代数、图像到坐标几何和应用题都会涉及。本复习指南把核心方法、关键知识点和常见考试陷阱集中在一处，以中英双语编写，帮助你建立准确的双语数学表达。
Work through the sections in order: each one gives the method, a worked example and the wording examiners actually use. Everything here is examinable, and everything here can be checked - so keep a pencil and a calculator beside you as you read.
建议按顺序阅读每一节：每节都给出方法、例题以及考官实际使用的表述。这里的内容都属于考试范围，也都可以自行验算，所以阅读时请准备好铅笔和计算器。
1. What is a Quadratic Equation? | 什么是二次方程？
numbers and a ≠ 0. If a were zero, the equation would become linear rather than quadratic.
二次方程是变量最高次数为 2 的多项式方程。其一般形式为 ax² + bx + c = 0，其中 a、b、c 为实数且 a ≠ 0。如果 a 为零，方程就变成一次方程而不是二次方程。
ax² + bx + c = 0 (a ≠ 0)
x² + 4x = 0 and 3x² − 7 = 0, even though they do not show every term explicitly. The word 'quadratic' comes from the Latin word 'quadratus', meaning square, because the variable is squared.
例如，x² − 5x + 6 = 0 和 2x² + 3x − 2 = 0 都是二次方程。x² = 0、x² + 4x = 0 和 3x² − 7 = 0 也属于二次方程，尽管它们没有明确写出每一项。'Quadratic' 一词源自拉丁语 'quadratus'，意为平方，因为方程中的变量被平方。
In IGCSE exams you may see quadratic equations written in many different forms, often hidden inside a word problem or a geometry question. You should be able to recognise them quickly and rearrange them into standard form before solving. Recognising the degree of the equation is the first step to choosing the correct method.
在 IGCSE 考试中，二次方程会以多种不同形式出现，常常隐藏在应用题或几何题里。你应当能够快速识别它们，并在求解前整理成标准形式。识别方程的次数是选择正确方法的第一步。
2. Standard Form and Coefficients | 标准形式与系数
c term, bx is the linear term and c is the constant term. The coefficient a may be positive or negative, but it must not be zero.
在求解之前，总是先把方程整理成标准形式 ax² + bx + c = 0。ax² 项称为二次项，bx 项称为一次项，c 是常数项。系数 a 可以为正或为负，但不能为零。
written as −2x² − 3x + 7 = 0, which is equivalent to 2x² + 3x − 7 = 0 after multiplying by −1. Changing every sign does not change the solutions.
整理方程能帮助你正确识别 a、b、c，以便进行因式分解或代入公式。例如，7 − 3x = 2x² 应先写成 −2x² − 3x + 7 = 0，等价于乘以 −1 后的 2x² + 3x − 7 = 0。改变所有项的符号不会改变方程的解。
nd c = −8. This habit prevents sign errors when you substitute into the quadratic formula.
把缺失的项写成零也很有帮助。例如，2x² − 8 = 0 可以看作 2x² + 0x − 8 = 0，因此 a = 2、b = 0、c = −8。养成这个习惯可以避免代入求根公式时出现符号错误。
3. Solving by Factorisation | 用因式分解求解
is zero, then at least one factor must be zero. In symbols, if pq = 0 then p = 0 or q = 0.
当根为有理数时，因式分解是最快的方法。它依据零积性质：若两个因式的乘积为零，则至少有一个因式为零。用符号表示即：若 pq = 0，则 p = 0 或 q = 0。
= 0, because (−2) × (−3) = 6 and (−2) + (−3) = −5. The roots are therefore x = 2 or x = 3.
对于 x² + bx + c，要找两个数，使它们的乘积为 c、和为 b。例如，x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0，因为 (−2) × (−3) = 6 且 (−2) + (−3) = −5。因此根为 x = 2 或 x = 3。
−x give 2x² + 4x − x − 2 = 2x(x + 2) − 1(x + 2) = (2x − 1)(x + 2). Hence x = ½ or x = −2.
当 a 不为 1 时，找出乘积为 ac、和为 b 的两个数，再拆分中间项。在 2x² + 3x − 2 = 0 中，ac = −4，用 +4x 和 −x 可得 2x² + 4x − x − 2 = 2x(x + 2) − 1(x + 2) = (2x − 1)(x + 2)。因此 x = ½ 或 x = −2。
- Write the equation in standard form, with zero on the right-hand side. | 先把方程写成标准形式，右边为零。
- Factorise completely, taking out any common factor first. | 完整分解，先提取公因式。
- Set each bracket equal to zero and solve. | 令每个括号等于零并求解。
- Check both roots by substituting back into the original equation. | 把两个根代回原方程检验。
4. Completing the Square | 配方法
bx = (x + b/2)² − (b/2)², so you add and subtract the square of half the coefficient of x.
配方法把二次式改写为 (x + p)² + q 的形式，使图像的顶点一目了然。核心恒等式是 x² + bx = (x + b/2)² − (b/2)²，也就是加上并减去一次项系数一半的平方。
0 becomes (x + 3)² − 9 + 1 = 0, that is (x + 3)² = 8. Taking square roots gives x + 3 = ±√8, so x = −3 ± 2√2, which is approximately x ≈ −0.17 or x ≈ −5.83.
例如，x² + 6x + 1 = 0 可写成 (x + 3)² − 9 + 1 = 0，即 (x + 3)² = 8。开平方得 x + 3 = ±√8，因此 x = −3 ± 2√2，约等于 x ≈ −0.17 或 x ≈ −5.83。
)² − 4] + 3 = 2(x + 2)² − 5. The minimum value of the expression is −5, reached at x = −2.
当 a ≠ 1 时，先从前两项中提出系数 a，再配方：2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。该式的最小值为 −5，在 x = −2 处取得。
5. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation, including those that will not factorise. Substitute the values of a, b and c from the standard form carefully, using brackets for negative numbers.
求根公式可以解任何二次方程，包括无法因式分解的方程。把标准形式中的 a、b、c 仔细代入，遇到负数时要加上括号。
x = (−b ± √b² − 4ac) / 2a
² + 3x − 7 = 0, where a = 2, b = 3 and c = −7. Then b² − 4ac = 9 + 56 = 65, so x = (−3 ± √
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