📚 Quadratic Equations: An IGCSE Teacher’s Guide (G-2, Unit 900) | 二次方程：IGCSE教师指南（G-2，第900单元）
This guide is designed for IGCSE Mathematics teachers and students. It provides a structured, exam-focused review of quadratic equations, covering standard forms, solution methods, graphing, and common pitfalls. The content aligns with the G-2 curriculum sequence and supports effective classroom delivery.
本指南专为IGCSE数学教师和学生编写，系统梳理二次方程的核心知识，包括标准形式、解法、图象以及常见易错点。内容紧扣G-2教学进度，帮助教师高效授课、学生精准备考。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2. Its general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The variable x represents an unknown number.
二次方程是最高次项为2的多项式方程，其一般形式为 ax² + bx + c = 0，其中a、b、c为常数，且a ≠ 0。变量x表示未知数。
For example, 2x² – 5x + 3 = 0 is quadratic, while x² – 4x + 1 = 0 is also quadratic. If a = 0, the equation becomes linear and is no longer quadratic.
例如，2x² – 5x + 3 = 0是二次方程，x² – 4x + 1 = 0也是二次方程。若a = 0，方程退化为一次方程，就不再是二次方程。
2. Solving by Factorisation | 用因式分解法解方程
Factoring is often the fastest method when the quadratic expression can be written as a product of two linear binomials. For example, to solve x² + 5x + 6 = 0, we factor it as (x + 2)(x + 3) = 0.
当二次式能写成两个一次二项式的乘积时，因式分解法通常是最快捷的解法。例如，解x² + 5x + 6 = 0，可因式分解为(x + 2)(x + 3) = 0。
Then apply the zero product property: if AB = 0, then A = 0 or B = 0. Therefore x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
然后利用零乘积性质：若AB = 0，则A = 0或B = 0。因此x + 2 = 0或x + 3 = 0，解得x = -2或x = -3。
- Always rearrange the equation to the form ax² + bx + c = 0 first.
- 先整理成ax² + bx + c = 0的形式。
- Factorise the quadratic expression completely.
- 将二次式彻底因式分解。
- Set each factor equal to zero and solve for x.
- 令每个因式等于零并解出x。
3. Solving by the Quadratic Formula | 用求根公式解方程
The quadratic formula works for any quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0, the solutions are given by:
求根公式适用于任何二次方程，即使因式分解困难或无法分解时也能求解。对于ax² + bx + c = 0，解为：
x = (-b ± √(b² – 4ac)) / (2a)
Here the symbol ± means that there are two solutions: one with a plus sign and one with a minus sign. For example, for x² – 3x – 4 = 0, we have a = 1, b = -3, c = -4.
这里的±表示有两个解：一个取加号，一个取减号。例如，对于x² – 3x – 4 = 0，有a = 1，b = -3，c = -4。
Substitute into the formula: x = (3 ± √(9 + 16)) / 2 = (3 ± 5) / 2, giving x = 4 or x = -1.
代入公式：x = (3 ± √(9 + 16)) / 2 = (3 ± 5) / 2，得到x = 4或x = -1。
4. Solving by Completing the Square | 用配方法解方程
Completing the square rewrites a quadratic in the form a(x + p)² + q = 0. This method is useful for deriving the quadratic formula and for solving equations where factorisation is not obvious.
配方法将二次式改写为a(x + p)² + q = 0的形式。此方法可用于推导求根公式，也适用于因式分解不明显时的求解。
Example: solve x² + 6x + 2 = 0. Take the coefficient of x, which is 6, halve it to get 3, then square it to get 9. Write x² + 6x = (x + 3)² – 9.
示例：解x² + 6x + 2 = 0。取x项的系数6，一半为3，平方得9。于是x² + 6x = (x + 3)² – 9。
Therefore (x + 3)² – 9 + 2 = 0, so (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, hence x = -3 ± √7.
因此(x + 3)² – 9 + 2 = 0，即(x + 3)² = 7。开平方得x + 3 = ±√7，所以x = -3 ± √7。
5. The Discriminant and Nature of Roots | 判别式与根的性质
The expression b² – 4ac is called the discriminant, often denoted by Δ. The value of Δ tells us the number and type of roots without solving the equation completely.
表达式b² – 4ac称为判别式，常用Δ表示。Δ的值可以判断方程根的情况，而无需完整求解方程。
- If Δ > 0, there are two distinct real roots.
- 若Δ > 0，有两个不相等的实数根。
- If Δ = 0, there is one repeated real root (a double root).
- 若Δ = 0，有一个重根（两个相等的实数根）。
- If Δ < 0, there are no real roots (two complex roots).
- 若Δ < 0，没有实数根（有两个复数根）。
For example, the equation 2x² + 3x + 5 = 0 has Δ = 9 – 40 = -31, so it has no real roots.
例如，方程2x² + 3x + 5 = 0的Δ = 9 – 40 = -31，因此没有实数根。
6. Sum and Product of Roots | 根的和与积
For a quadratic equation ax² + bx + c = 0, if the roots are α and β, then the sum and product of the roots relate directly to the coefficients:
对于二次方程ax² + bx + c = 0，若两根为α和β，则根的和与积直接与系数相关：
α + β = -b/a, αβ = c/a
These relations are useful for checking solutions and for constructing equations with given roots.
这些关系可用于检验解，也可用于构造具有指定根的二次方程。
For instance, if the roots of x² – 7x + 12 = 0 are 3 and 4, then their sum is 7 and product is 12, matching the formula.
例如，x² – 7x + 12 = 0的两根为3和4，则根和为7，根积为12，与公式一致。
7. Graphs of Quadratic Functions | 二次函数的图象
The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upward (U-shape). When a < 0, it opens downward (n-shape).
y = ax² + bx + c的图象是一条抛物线。当a > 0时，抛物线开口朝上（U形）；当a < 0时，开口朝下（n形）。
The roots of the equation ax² + bx + c = 0 are the x-coordinates of the points where the parabola crosses the x-axis. If Δ = 0, the vertex touches the x-axis at one point; if Δ < 0, the parabola does not cross the x-axis at all.
方程ax² + bx + c = 0的根，是抛物线与x轴交点的x坐标。若Δ = 0，抛物线的顶点与x轴相切；若Δ < 0，则抛物线与x轴没有交点。
The vertex of the parabola has x-coordinate x = -b/(2a). The axis of symmetry is the vertical line through the vertex.
抛物线的顶点横坐标为x = -b/(2a)，对称轴是经过顶点的竖直线。
8. Solving Word Problems | 解应用题
Many real-world problems can be modelled by quadratic equations. A typical approach involves defining a variable, setting up an equation from the given conditions, and solving it.
许多实际问题可以用二次方程建模。典型步骤是：设未知数，根据条件列出方程，然后求解。
Example: The area of a rectangle is 36 cm², and its length is 5 cm longer than its width. Find the width.
示例：一个矩形的面积为36 cm²，长比宽长5 cm。求宽是多少？
Let the width be x cm. Then the length is x + 5 cm. The area equation is x(x + 5) = 36, which expands to x² + 5x – 36 = 0. Factorise: (x + 9)(x – 4) = 0, so x = -9 or x = 4. Since a width cannot be negative, the width is 4 cm.
设宽为x cm，则长为(x + 5) cm。面积方程为x(x + 5) = 36，展开为x² + 5x – 36 = 0。因式分解：(x + 9)(x – 4) = 0，所以x = -9或x = 4。宽度不能为负，因此宽为4 cm。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Students often make sign errors when substituting coefficients into the quadratic formula. Write down the values of a, b, and c clearly before substituting.
学生在将系数代入求根公式时常犯符号错误。代入前应清楚地写出a、b、c的值。
Another common mistake is forgetting to set the equation to zero before factorising. Always move all terms to one side first.
另一个常见错误是忘记在因式分解前将方程化为零。应先将所有项移到等式一侧。
When solving by taking square roots, remember that square roots can be positive or negative. For instance, if x² = 9, then x = ±3, not just 3.
用开平方法求解时，记住平方根有正负两个值。例如，x² = 9，则x = ±3，而不仅仅是3。
10. Practice Exercises and Teaching Tips | 练习与教学提示
The G-2 teacher’s manual contains 900 carefully graded exercises. Teachers should begin with simple factorisation, then move to the quadratic formula, and finally to word problems and graphing.
G-2教师用书包含900道精心分级的练习题。教师应从简单的因式分解开始，再过渡到求根公式，最后再涉及应用题和图象。
| Topic | 主题 | Exercises | 题数 |
|---|---|
| Factorisation | 因式分解 | 300 |
| Quadratic formula | 求根公式 | 250 |
| Completing the square | 配方法 | 150 |
| Word problems | 应用题 | 120 |
| Graphs and discriminant | 图象与判别式 | 80 |
Encourage students to verify their answers by substituting back into the original equation. Use graphical calculators or online tools to explore how changing a, b, and c affects the parabola.
鼓励学生将解代回原方程进行验证。也可以使用图形计算器或在线工具，探索a、b、c的变化如何影响抛物线。
11. Summary of Key Formulae | 关键公式总结
The following list contains every essential formula for solving quadratic equations in IGCSE mathematics. Keep a copy for revision before the exam.
以下列出IGCSE数学中解二次方程所需的所有关键公式。考前复习请保留此清单。
- General form: ax² + bx + c = 0
- 一般形式：ax² + bx + c = 0
- Quadratic formula: x = (-b ± √(b² – 4ac)) / (2a)
- 求根公式：x = (-b ± √(b² – 4ac)) / (2a)
- Discriminant: Δ = b² – 4ac
- 判别式：Δ = b² – 4ac
- Sum of roots: α + β = -b/a
- 根之和：α + β = -b/a
- Product of roots: αβ = c/a
- 根之积：αβ = c/a
- Vertex x-coordinate: x = -b/(2a)
- 顶点横坐标：x = -b/(2a)
12. Final Advice for Teachers | 给教师的最终建议
Quadratic equations are a cornerstone of IGCSE algebra. Students who master this topic build a strong foundation for calculus, trigonometry, and coordinate geometry.
二次方程是IGCSE代数的基石。学生掌握好这一主题，能为微积分、三角学和坐标几何打下坚实的基础。
Use the 900 exercises in the G-2 teacher’s manual flexibly: assign shorter sets for daily practice, mix question types in weekly quizzes, and reserve longer word problems for group work or homework projects.
灵活使用G-2教师用书中的900道练习题：可布置短小练习用于每日训练，在每周测验中混合不同题型，并将较长的应用题保留为小组合作或课外作业。
Always stress the meaning behind equations: roots are the x-values where the graph crosses the x-axis, and the discriminant tells us how many times it crosses. Connect algebra to geometry to deepen understanding.
始终强调方程背后的几何意义：根是图象与x轴交点的横坐标，判别式告诉我们交点个数。将代数与图形联系起来，可以加深理解。
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