📚 IGCSE Maths Grade 6 Topic Series 100: Quadratic Equations | IGCSE 数学 6 级主题系列 100：二次方程
Quadratic equations appear throughout the IGCSE syllabus and are one of the most common sources of marks in algebra papers. This topic guide covers everything you need to reach a secure Grade 6 level, from recognising a quadratic to factorising, using the formula, sketching graphs, and solving real-world problems.
二次方程贯穿 IGCSE 数学大纲，是代数试卷中最常见的得分来源之一。本专题指南涵盖从识别二次方程、因式分解、使用求根公式、绘制函数图像到解决实际问题等所有内容，帮助你达到 6 级水平。
1. What is a Quadratic Equation? | 什么是二次方程？
≠ 0. The key feature is the x² term; if a = 0 the equation becomes linear, not quadratic.
一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数且 a ≠ 0。关键特征是含有 x² 项；如果 a = 0，方程就变成了一次方程，不再是二次方程。
Quadratic comes from the Latin word 'quadratus' meaning square. The unknown x is raised to the power 2, which is why the equation is called quadratic.
“二次”一词来源于拉丁语“quadratus”，意思是平方。未知数 x 的指数为 2，因此这种方程被称为二次方程。
- Examples: x² + 5x + 6 = 0, 2x² − 3x − 2 = 0, x² − 9 = 0
- Non-examples: x³ + 2x = 0, 3x + 7 = 0, 1/x² + x = 0
- 示例：x² + 5x + 6 = 0，2x² − 3x − 2 = 0，x² − 9 = 0
- 非示例：x³ + 2x = 0，3x + 7 = 0，1/x² + x = 0
2. Standard Form and Coefficients | 标准形式与系数
coefficient a is the number multiplying x², b is the number multiplying x, and c is the constant term. Always rearrange an equation into this form before solving.
标准形式为 ax² + bx + c = 0。系数 a 是 x² 前的数，b 是 x 前的数，c 是常数项。在求解之前，一定要先把方程整理成这种形式。
x − 6 = 4, then subtract 4 from both sides: x² − x − 10 = 0. Here a = 1, b = −1, c = −10.
例如，展开并化简 (x + 2)(x − 3) = 4，得到 x² − x − 6 = 4，然后两边同时减去 4：x² − x − 10 = 0。此时 a = 1，b = −1，c = −10。
3. Solving by Factorising | 因式分解法
acket equal to zero. The method relies on the fact that if A × B = 0, then A = 0 or B = 0.
当二次式可以因式分解时，通过让每个括号等于零，就能快速解出方程。这种方法基于：如果 A × B = 0，那么 A = 0 或 B = 0。
Steps: write the equation in standard form, factorise the left-hand side, set each factor equal to zero, and solve the resulting linear equations.
步骤：把方程写成标准形式，对左边进行因式分解，令每个因式等于零，再解得到的一次方程。
= 0. Factorise to (x + 3)(x + 4) = 0, so x + 3 = 0 or x + 4 = 0, giving x = −3 or x = −4.
示例：解 x² + 7x + 12 = 0。因式分解得 (x + 3)(x + 4) = 0，所以 x + 3 = 0 或 x + 4 = 0，解得 x = −3 或 x = −4。
2x² + 6x − x − 3 = 0, then factor by grouping to (x + 3)(2x − 1) = 0, so x = −3 or x = ½.
当系数 a 不等于 1 时，可以用拆分中间项或尝试法。对于 2x² + 5x − 3 = 0，把 5x 拆成 6x − x：2x² + 6x − x − 3 = 0，再分组分解为 (x + 3)(2x − 1) = 0，所以 x = −3 或 x = ½。
4. Solving by Completing the Square | 配方法
Completing the square transforms x² + bx into a perfect square plus a constant. It is useful when factorising is difficult, and it leads directly to the quadratic formula.
配方法把 x² + bx 转化为完全平方加上一个常数。它在因式分解困难时很有用，并且可以直接推导出求根公式。
For x² + bx, add and subtract (b/2)²: x² + bx = (x + b/2)² − (b/2)².
对于 x² + bx，加减 (b/2)²：x² + bx = (x + b/2)² − (b/2)²。
d (6/2)² = 9 to both sides: x² + 6x + 9 = 7, so (x + 3)² = 7. Take square roots: x + 3 = ±√7, hence x = −3 ± √7.
示例：解 x² + 6x + 2 = 0。移项得 x² + 6x = −2。两边同时加上 (6/2)² = 9：x² + 6x + 9 = 7，即 (x + 3)² = 7。开平方：x + 3 = ±√7，所以 x = −3 ± √7。
5. The Quadratic Formula | 求根公式
The quadratic formula gives the solutions of ax² + bx + c = 0 as x = (−b ± √b² − 4ac) / 2a. It works for every quadratic equation, including those that cannot be factorised.
求根公式给出 ax² + bx + c = 0 的解：x = (−b ± √b² − 4ac) / 2a。它适用于所有二次方程，包括无法因式分解的方程。
x = (−b ± √b² − 4ac) / 2a
Remember to substitute a, b and c carefully, especially when they are negative. Use brackets around negative values to avoid sign errors.
代入 a、b、c 时要仔细，尤其是负数。负数要用括号括起来，以避免符号错误。
Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute: x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2.
示例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入：x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2。
6. The Discriminant | 判别式
ic formula: Δ = b² − 4ac. It tells you how many real solutions the quadratic equation has.
判别式是求根公式中平方根内的部分：Δ = b² − 4ac。它告诉你二次方程有多少个实数解。
- If Δ > 0, there are two distinct real solutions.
- If Δ = 0, there is exactly one real solution (a repeated root).
- If Δ < 0, there are no real solutions (two complex roots, not needed for IGCSE).
- 如果 Δ > 0，方程有两个不同的实数解。
- 如果 Δ = 0，方程只有一个实数解（重根）。
- 如果 Δ < 0，方程没有实数解（两个复数根，IGCSE 不要求）。
e: For x² + 4x + 5 = 0, Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4, so there are no real solutions.
示例：对于 x² + 4x + 5 = 0，Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4，所以没有实数解。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0 the parabola opens upwards (a minimum point); if a < 0 it opens downwards (a maximum point).
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上（有最低点）；如果 a < 0，抛物线开口向下
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