G-4: Solving Quadratic Equations – A Teacher’s Guide | G-4：解二次方程——教师用书指南

📚 G-4: Solving Quadratic Equations | G-4：解二次方程

Quadratic equations form one of the most critical topics in the IGCSE Mathematics syllabus. Students are expected to solve them using multiple methods, interpret their graphical meaning, and apply them to real-world problems. This teacher’s guide provides a structured approach to delivering this topic effectively in the classroom.

二次方程是IGCSE数学大纲中最关键的主题之一。学生需要掌握多种解法、理解其图像意义，并能将其应用于实际问题。本教师用书提供了一种结构化的教学方法，帮助教师高效地在课堂上传授这一主题。

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1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is an equation in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The coefficient a determines the curvature of the parabola, while b and c affect its position. It is essential that students can identify the coefficients quickly, as this forms the basis for every solving method.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。系数 a 决定抛物线的开口方向和宽窄，b 和 c 影响图像的位置。学生必须能够迅速识别各项系数，因为这是所有解法的基础。

For example, in the equation 2x² – 5x + 3 = 0, a = 2, b = -5, and c = 3. Written work should always include the clear identification of these values before any further steps are attempted.

例如，在方程 2x² – 5x + 3 = 0 中，a = 2，b = -5，c = 3。在尝试任何进一步步骤之前，学生的书写中应始终清楚标明这些值。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the first method taught because it emphasises the structure of the equation. The key is to rewrite ax² + bx + c as a product of two binomials. For example, x² + 5x + 6 factors as (x + 2)(x + 3). Setting each factor to zero gives x = -2 or x = -3.

因式分解通常是首先教授的方法，因为它强调方程的结构。关键是将 ax² + bx + c 重写为两个二项式的乘积。例如，x² + 5x + 6 可分解为 (x + 2)(x + 3)。令每个因子为零，得到 x = -2 或 x = -3。

Teachers should stress that this method only works when the equation has rational roots. Students must practice recognising perfect squares and difference of two squares, such as x² – 9 = (x + 3)(x – 3).

教师应强调，只有在方程有有理根时，因式分解法才有效。学生必须练习识别完全平方和平方差，例如 x² – 9 = (x + 3)(x – 3)。

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3. Completing the Square | 配方法

Completing the square transforms the quadratic into the form a(x – p)² + q. This is particularly useful for finding the turning point and for solving equations that do not factor. For instance, x² + 6x + 5 can be written as (x + 3)² – 4, leading to x = -3 ± 2, so x = -1 or x = -5.

配方法将二次方程转化为 a(x – p)² + q 的形式。这在求顶点坐标和解决无法因式分解的方程时特别有用。例如，x² + 6x + 5 可写成 (x + 3)² – 4，从而得到 x = -3 ± 2，所以 x = -1 或 x = -5。

Students often struggle with the required half-the-coefficient step. A helpful teaching tip is to always write out the constant term separately and adjust it carefully. Stress that the vertex form directly reveals the line of symmetry: x = p.

学生常常在“取一半系数”这一步遇到困难。一个有用的教学技巧是始终将常数项单独写出，并仔细调整。强调顶点式直接揭示对称轴：x = p。

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4. The Quadratic Formula | 二次公式

The quadratic formula works for all quadratics, factorable or not. The formula is:

x = (-b ± √(b² – 4ac)) / (2a)

Where a, b, and c are the coefficients from the standard form. This method should be used when factorisation is difficult or impossible, and it also prepares students for more advanced coordinate geometry.

二次公式适用于所有二次方程，无论是否可因式分解。公式为：

x = (-b ± √(b² – 4ac)) / (2a)

其中 a、b、c 是标准形式中的系数。当因式分解困难或不可能时，应使用此方法，同时也为学生学习更高阶的坐标几何做好准备。

Teachers should guide students to substitute the coefficients accurately and to simplify surds carefully. Many mark schemes reward intermediate steps, so encourage clear working rather than relying solely on a calculator.

教师应指导学生准确代入系数，并谨慎化简根式。许多评分标准都重视中间步骤，因此应鼓励学生写出清晰的演算过程，而不仅仅依赖计算器。

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5. The Discriminant and the Nature of Roots | 判别式与根的性质

The discriminant, Δ = b² – 4ac, determines the number and type of roots. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real roots, but two complex roots.

判别式 Δ = b² – 4ac 决定根的数量和类型。如果 Δ > 0，方程有两个不相等的实数根；如果 Δ = 0，方程有一个实数根（重根）；如果 Δ < 0，方程没有实数根，但有两个复数根。

This theme regularly appears in IGCSE exams either as a direct calculation or as a condition for a certain number of roots. For example, a question might ask: “Find the values of k for which x² + kx + 4 = 0 has equal roots.” Here students set Δ = 0 and solve k² – 16 = 0, giving k = ±4.

这个知识点在IGCSE考试中经常出现，要么直接计算判别式，要么作为某个根数目的条件。例如，题目可能问：“求 x² + kx + 4 = 0 有等根时 k 的值。”学生需要令 Δ = 0，然后解 k² – 16 = 0，得到 k = ±4。

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6. Graphs of Quadratic Functions | 二次函数图像

A quadratic function y = ax² + bx + c produces a parabola. The sign of a tells us whether it opens upwards (a > 0) or downwards (a < 0). The roots are the x-intercepts, and the axis of symmetry lies at x = -b/(2a).

二次函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定开口方向（a > 0 开口向上，a < 0 开口向下）。根就是与 x 轴的交点，对称轴位于 x = -b/(2a)。

Students should also be able to find the y-intercept by setting x = 0, and to locate the vertex using completing the square or the formula for the line of symmetry. Sketching a quick graph helps verify the number of roots.

学生还应能够通过令 x = 0 求 y 轴截距，并通过配方法或对称轴公式求顶点坐标。快速画一个草图有助于验证根的个数。

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7. Application Problems | 应用题

Quadratic equations arise naturally in geometry, movement, and economics. A typical problem might involve finding the dimensions of a rectangle given its area and perimeter. If a rectangle has length x and width (10 – x), and the area is 24, the equation becomes x(10 – x) = 24, which simplifies to x² – 10x + 24 = 0.

二次方程在几何、运动和经济问题中自然出现。典型题目可能是在已知面积和周长的情况下求矩形的尺寸。若矩形的长为 x，宽为 (10 – x)，面积为 24，则方程为 x(10 – x) = 24，化简得 x² – 10x + 24 = 0。

Encourage students to define variables clearly, form the equation, solve it, and then reject any solution that is impossible in context. For example, a negative length should be discarded even if it satisfies the algebra.

鼓励学生清楚定义变量，列出方程，求解，然后排除不符合实际意义的解。例如，负数长度即使在代数上成立也应舍弃。

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8. Common Mistakes and Teaching Strategies | 常见错误与教学策略

Common errors include forgetting to write the equation in standard form, mixing up signs when substituting into the quadratic formula, and omitting the ± symbol. Another frequent issue is attempting factorisation when the quadratic does not factor over integers.

常见错误包括忘记将方程写成标准形式、代入二次公式时弄错符号、遗漏 ± 符号。另一个常见问题是当二次方程无法用整数因式分解时仍强行使用因式分解法。

Effective teaching strategies include:

有效的教学策略包括：

• Always have students check that a ≠ 0 before identifying coefficients.

  让学生在识别系数前总是先检查 a ≠ 0。

• Use colour-coding for a, b, and c when working through examples.

  在演算例题时使用不同颜色标记 a、b、c。

• Provide plenty of practice with the discriminant to avoid blind reliance on factorisation.

  提供大量关于判别式的练习，避免盲目依赖因式分解。

• Show both algebraic and graphical solutions to reinforce the same concept.

  同时展示代数解法和图像解法，以强化同一概念。

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9. Revision and Assessment | 复习与评估

For revision, create a summary table of the three main methods—factorisation, completing the square, and the quadratic formula—listing when to use each. A short diagnostic test after teaching each subsection helps identify weak areas early.

复习时，可以制作一个三种主要方法（因式分解、配方法、二次公式）的总结表，列出各自适用的情况。在教授每个小节后进行简短诊断测试，有助于尽早发现薄弱环节。

Method	Best Used When
Factorisation	Rational roots visible quickly
Completing the square	Turning point needed
Quadratic formula	Any quadratic, especially non-factorable

Assessments should include non-calculator questions to test algebraic fluency, as well as calculator questions for more complex surd simplifications. Past-paper style questions are invaluable for building exam technique.

评估应包括不使用计算器的题目以测试代数熟练度，以及使用计算器处理复杂根式化简的题目。真题风格的题目对培养考试技巧非常宝贵。

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