📚 IGCSE Maths G-6 Teacher's Book 500: Quadratic Equations | IGCSE数学G-6教师用书500：二次方程
This G-6 Teacher's Book 500 unit focuses on quadratic equations, one of the most examined topics in IGCSE Mathematics. Students need to be confident when moving between standard form, factorised form, completed-square form, and the graph of a quadratic function.
本单元 G-6 教师用书 500 聚焦二次方程，这是 IGCSE 数学中最常考查的主题之一。学生需要熟练地在标准式、因式分解式、配方式以及二次函数图像之间进行转换。
1. What is a Quadratic Equation? | 什么是二次方程？
, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential: if a were zero, the x² term would disappear and the equation would become linear.
在 IGCSE 数学中，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。a ≠ 0 这个条件非常关键：如果 a 等于 0，x² 项就会消失，方程就变成一次方程。
Common examples include:
常见例子包括：
- 3x² + 2x − 5 = 0
- x² − 16 = 0
- 2x² = 7x + 4
The last example must be rearranged into standard form before solving: 2x² − 7x − 4 = 0.
最后一个例子在求解前需要先整理成标准形式：2x² − 7x − 4 = 0。
2. Solving by Factorising | 因式分解法求解
o. For example, x² + 5x + 6 = 0 factorises as (x + 2)(x + 3) = 0, giving x = −2 or x = −3.
当二次方程具有简单因式时，因式分解法通常是最快的方法。先把方程写成标准形式，对二次式进行因式分解，然后令每个因式等于零。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0，因此 x = −2 或 x = −3。
x² + 5x + 6 = 0 ⇒ (x + 2)(x + 3) = 0 ⇒ x = −2 or x = −3
Always check that the equation is equal to zero before factorising. A common mistake is to factorise the left-hand side while the right-hand side is still a number other than zero.
在进行因式分解前，一定要先确认方程右边为零。一个常见错误是：右边还不是零时，就提前对左边进行了因式分解。
3. Completing the Square | 配方法
g equations and for finding the vertex. The key identity is x² + bx = (x + b/2)² − (b/2)².
配方法可以把二次式转化为 a(x − h)² + k 的形式，这种形式对解方程和求顶点都非常有用。核心恒等式是 x² + bx = (x + b/2)² − (b/2)²。
x² + 10x + 7 = 0 becomes (x + 5)² − 25 + 7 = 0, so (x + 5)² = 18, and therefore x = −5 ± 3√2.
例如，x² + 10x + 7 = 0 可以写成 (x + 5)² − 25 + 7 = 0，所以 (x + 5)² = 18，因此 x = −5 ± 3√2。
x² + 10x + 7 = 0 ⇒ (x + 5)² − 18 = 0 ⇒ x = −5 ± 3√2
This method is especially important when the roots are not integers or when the question asks for exact answers in surd form.
当根不是整数，或者题目要求用根式给出精确答案时，配方法尤其重要。
4. The Quadratic Formula | 二次公式法
ing is difficult or impossible. For ax² + bx + c = 0, the solutions are given by x = (−b ± √b² − 4ac) / (2a).
二次公式可以求解任何二次方程，即使因式分解很难或无法进行。对于 ax² + bx + c = 0，解的公式为 x = (−b ± √b² − 4ac) / (2a)。
x = (−b ± √b² − 4ac) / (2a)
carefully. Pay close attention to negative signs: if b = −3, then −b = +3 in the formula.
使用公式前，要先把方程整理成标准形式，并仔细辨认 a、b、c。特别注意负号：如果 b = −3，那么公式中的 −b 就是 +3。
For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3, so x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2.
例如，求解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3，所以 x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2。
5. Discriminant and Nature of Roots | 判别式与根的性质
ula: Δ = b² − 4ac. It tells us the nature of the roots without solving the equation fully.
判别式是二次公式中平方根号下的部分：Δ = b² − 4ac。它可以在不完整求解方程的情况下判断根的性质。
| Discriminant | Nature of roots | Graph behaviour |
|---|---|---|
| Δ > 0 | Two distinct real roots 两个不相等实根 |
Parabola crosses x-axis twice 抛物线与x轴相交两次 |
| Δ = 0 | One repeated real root 一个重复实根 |
Parabola touches x-axis once 抛物线与x轴相切一次 |
| Δ < 0 | No real roots 没有实根 |
Parabola does not cut x-axis 抛物线与x轴不相交 |
IGCSE questions often ask for the set of values of k for which an equation has two distinct real roots, equal roots, or no real roots. These problems are solved by writing an inequality or equation involving Δ.
IGCSE 题目经常要求找出使方程有两个不相等实根、相等实根或没有实根的 k 值范围。这类问题需要通过列出关于 Δ 的不等式或方程来求解。
6. Graphs of Quadratic Functions | 二次函数图像
0, it opens downwards like an inverted U. The y-intercept is c, because when x = 0, y = c.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，呈 U 形；如果 a < 0，抛物线开口向下，呈倒 U 形。y 轴截距是 c，因为当 x = 0 时，y = c。
criminant is positive, there are two x-intercepts; if zero, one x-intercept; if negative, none.
x 轴截距就是方程 ax² + bx + c = 0 的实根。如果判别式为正，则有两个 x 轴截距；如果为零，则有一个 x 轴截距；如果为负，则没有 x 轴截距。
Sketching a quadratic graph does not require plotting many points. You only need to identify the shape, the y-intercept, the roots if they exist, and the vertex.
画二次函数草图不需要描出许多点。你只需要确定开口方向、y 轴截距、存在的根以及顶点即可。
7. Vertex and Axis of Symmetry | 顶点与对称轴
tex lies on this line, so its x-coordinate is −b / (2a), and its y-coordinate is found by substituting this value into the function.
y = ax² + bx + c 的对称轴是竖直线 x = −b / (2a)。顶点就在这条直线上，所以顶点的 x 坐标为 −b / (2a)，y 坐标可以通过把这个值代入函数得到。
ately (h, k). For example, y = 2(x − 3)² + 4 has vertex (3, 4) and axis of symmetry x = 3.
如果二次函数写成了配方式 y = a(x − h)² + k，那么顶点可以直接读出为 (h, k)。例如，y = 2(x − 3)² + 4 的顶点是 (3, 4)，对称轴是 x = 3。
This link between algebra and graph is frequently tested: a question may give a graph and ask for the equation in completed-square form, or vice versa.
代数与图像之间的这种联系经常被考查：题目可能给出一条图像，要求写出配方式；也可能给出配方式，要求描述图像特征。
8. Solving Quadratic Inequalities | 二次不等式求解
sign table or a quick sketch shows that the product is positive when x < 2 or x > 3.
要解二次不等式，例如 x² − 5x + 6 > 0，首先进行因式分解，得到 (x − 2)(x − 3) > 0。临界值为 x = 2 和 x = 3。通过符号表或快速草图可以判断，当 x < 2 或 x > 3 时，乘积为正。
x² − 5x + 6 > 0 ⇒ (x − 2)(x − 3) > 0 ⇒ x < 2 or x > 3
er the inequality includes equality, because this determines whether the endpoints are included or excluded.
如果不等式是 x² − 5x + 6 ≤ 0，解就是 2 ≤ x ≤ 3。一定要检查不等式是否包含等号，因为这会决定端点是否包含在解集中。
9. Applications and Word Problems | 应用与文字题
Quadratic equations often appear in real-life IGCSE problems such as area, product of numbers, and projectile height. Translate the words into an equation, solve it, and then decide which root makes sense in context.
二次方程经常出现在 IGCSE 的实际问题中，例如面积问题、数字乘积问题和抛射高度问题。先把文字转化为方程，求解后，再根据题意判断哪个根是合理的。
72 = 0. This factorises as (n + 9)(n − 8) = 0, giving n = 8, since n = −9 is not positive.
例子：两个连续正整数的乘积为 72。设较小的整数为 n。那么 n(n + 1) = 72，所以 n² + n − 72 = 0。因式分解得 (n + 9)(n − 8) = 0，因为 n = −9 不是正数，所以 n = 8。
When solving word problems, negative roots or
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