📚 Teacher's Book 600: Mastering Quadratic Equations | 教师用书600：掌握二次方程
Teacher's Book 600 is a focused IGCSE Mathematics unit that builds confidence in quadratic equations, quadratic functions, and their graphs. The resource is designed for classroom teaching and independent revision, with clear worked examples and exam-style practice.
教师用书600 是一个聚焦的 IGCSE 数学单元，旨在建立学生对二次方程、二次函数及其图像的信心。该资源适用于课堂教学和自主复习，配有清晰的例题和考试风格练习。
1. Why Teacher's Book 600 Matters | 为什么教师用书600很重要
The unit targets the core IGCSE algebra skills that appear every year in Paper 2 and Paper 4. Quadratic equations link factorising, completing the square, the quadratic formula, and graphical methods, so they test several skills at once.
该单元针对 IGCSE 数学卷二和卷四每年都会出现的核心代数技能。二次方程同时考查因式分解、配方法、求根公式和图像方法，因此能综合检验多种能力。
Teachers can use this unit as a diagnostic tool: if a student struggles with quadratics, it often reveals gaps in earlier algebra such as expanding brackets or manipulating negative numbers.
教师可以将本单元用作诊断工具：如果学生在二次方程上遇到困难，通常能反映出早期代数中的漏洞，例如展开括号或处理负数。
The structure moves from simple factorising to applied problems, which helps learners build both fluency and understanding.
其结构从简单因式分解逐步过渡到应用题，有助于学习者同时提高熟练度和理解力。
Beyond pure algebra, quadratics connect to coordinate geometry, graph transformations, inequalities, and optimisation. Mastering the topic therefore strengthens performance across many areas of IGCSE Mathematics.
除纯代数之外，二次式还与坐标几何、图像变换、不等式和最优化相联系。因此掌握该主题能提升 IGCSE 数学多个领域的表现。
2. Core Definitions and Notation | 核心定义与符号
a, b, and c are constants, and a ≠ 0. The related quadratic equation is ax² + bx + c = 0.
二次式是形如 ax² + bx + c 的表达式，其中 a、b、c 为常数，且 a ≠ 0。对应的二次方程为 ax² + bx + c = 0。
ax² + bx + c = 0, a ≠ 0
the quadratic is called monic. A root, zero, or solution is a value of x that makes the equation true.
系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。当 a = 1 时，该二次式称为首一二次式。根、零点或解是使方程成立的 x 值。
- Standard form: ax² + bx + c = 0, with all terms on one side.
- Root: a value that satisfies the equation.
- Discriminant: Δ = b² - 4ac.
- Vertex: the turning point of the parabola.
- Axis of symmetry: the vertical line x = -b ÷ (2a).
- 标准形式：ax² + bx + c = 0，所有项在等号一侧。
- 根：使方程成立的数值。
- 判别式：Δ = b² - 4ac。
- 顶点：抛物线的转折点。
- 对称轴：竖直线 x = -b ÷ (2a)。
3. Factorising Quadratics | 因式分解二次式
Factorising reverses expansion. For a monic quadratic x² + bx + c, find two numbers that multiply to c and add to b. Then write the expression as (x + p)(x + q).
因式分解是展开的逆过程。对于首一二次式 x² + bx + c，找到两个数，使其乘积为 c、和为 b，然后写成 (x + p)(x + q)。
x² + bx + c = (x + p)(x + q), pq = c, p + q = b
For a non-monic quadratic ax² + bx + c with a ≠ 1, split the middle term using two numbers that multiply to ac and add to b, then factor by grouping. Alternatively, use trial and error with factors of a and c.
对于 a ≠ 1 的非首一二次式 ax² + bx + c，用两个数拆分中间项，这两个数的乘积为 ac、和为 b，然后分组分解。也可以用 a 和 c 的因数进行试算。
f two squares, x² - k² = (x - k)(x + k), and the perfect square, x² + 2kx + k² = (x + k)².
考试中经常出现两种特殊模式：平方差 x² - k² = (x - k)(x + k)，以及完全平方 x² + 2kx + k² = (x + k)²。
(x - 2)(x - 3). Example: 2x² + 7x + 3 = (2x + 1)(x + 3). Always expand mentally to check.
例：x² - 5x + 6 = (x - 2)(x - 3)。例：2x² + 7x + 3 = (2x + 1)(x + 3)。务必在脑中展开检验。
4. Completing the Square | 配方法
bx + c, take half of b, square it, and adjust: x² + bx + c = (x + b ÷ 2)² - (b ÷ 2)² + c.
配方法将二次式改写为顶点形式。对于 x² + bx + c，取 b 的一半，平方后调整：x² + bx + c = (x + b ÷ 2)² - (b ÷ 2)² + c。
x² + bx + c = (x + b ÷ 2)² - (b ÷ 2)² + c
When a ≠ 1, first factor a out of the first two terms, complete the square inside the bracket, then multiply back. The completed-square form is a(x - h)² + k, where (h, k) is the vertex.
当 a ≠ 1 时，先从 前两项 中提取 a，在括号内配方，再乘回去。配方后的形式为 a(x - h)² + k，其中 (h, k) 是顶点。
ample: x² + 6x + 1 = (x + 3)² - 9 + 1 = (x + 3)² - 8. The minimum value is -8 when x = -3.
例：x² + 6x + 1 = (x + 3)² - 9 + 1 = (x + 3)² - 8。当 x = -3 时，最小值为 -8。
Completing the square is especially useful for finding maximum and minimum values, solving equations without the formula, and proving that a quadratic is always positive or always negative.
配方法特别适用于求最大值和最小值、不用公式解方程，以及证明二次式恒为正或恒为负。
5. The Quadratic Formula | 二次求根公式
ard form. It is derived by completing the square on the general equation ax² + bx + c = 0.
二次求根公式可解任何标准形式的二次方程。它是对一般方程 ax² + bx + c = 0 配方推导得到的。
x = ( -b ± √b² - 4ac ) ÷ (2a)
Before substituting, write the equation in standard form and identify a, b, and c carefully. Pay special attention to negative signs.
代入前，先将方程写成标准形式，并仔细确定 a、b、c。要特别注意负号。
- Step 1: Rearrange to ax² + bx + c = 0.
- Step 2: Write down a, b, and c.
- Step 3: Compute b² - 4ac.
- Step 4: Substitute into the formula.
- Step 5: Simplify the surd and check the answer.
- 步骤 1：整理为 ax² + bx + c = 0。
- 步骤 2：写出 a、b、c。
- 步骤 3：计算 b² - 4ac。
- 步骤 4：代入公式。
- 步骤 5：化简根式并检验答案。
Example: 2x² - 4x - 3 = 0. Here a = 2, b = -4, c = -3. Then x = (4 ± √16 + 24) ÷ 4 = (4 ± √40) ÷ 4 = 1 ± √10 ÷ 2.
例：2x² - 4x - 3 = 0。这里 a = 2，b = -4，c = -3。于是 x = (4 ± √16 + 24) ÷ 4 = (4 ± √40) ÷ 4 = 1 ± √10 ÷ 2。
6. The Discriminant and Number of Roots | 判别式与根的个数
- 4ac. It tells us how many real roots a quadratic equation has without solving it fully.
判别式为 Δ = b² - 4ac。它能在不完整求解的情况下告诉我们二次方程有多少个实根。
| Condition | Number of real roots | Graph behaviour |
|---|---|---|
| Δ > 0 | Two distinct real roots | Crosses the x-axis twice |
| Δ = 0 | One repeated real root | Touches the x-axis once |
| Δ < 0 | No real roots | Does not meet the x-axis |
中文对照：Δ > 0 时有两个不同实根，图像与 x 轴交于两点；Δ = 0 时有一个重根，图像与 x 轴相切于一点；Δ < 0 时没有实根，图像不与 x 轴相交。
Exams often ask for the range of values of k for which an equation has two real roots. Set Δ > 0 and solve the resulting inequality.
考试常问：当 k 取什么范围时方程有两个实根。令 Δ > 0 并解所得不等式。
Example: x² + kx + 9 = 0 has equal roots. Then Δ = k² - 36 = 0, so k = 6 or k = -6.
例：x² + kx + 9 = 0 有等根。则 Δ = k² - 36 = 0，所以 k = 6 或 k = -6。
7. Quadratic Graphs and Key Features | 二次函数图像与关键特征
. If a > 0, it opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.
y = ax² + bx + c 的图像是抛物线。若 a > 0，开口向上，有最低点；若 a < 0，开口向下，有最高点。
ax² + bx + c = 0. The axis of symmetry is x = -b ÷ (2a), and the vertex lies on this line.
y 轴截距为 c，令 x = 0 得到。x 轴截距（若存在）是 ax² + bx + c = 0 的根。对称轴为 x = -b ÷ (2a)，顶点在该直线上。
Vertex x-coordinate: x = -b ÷ (2a)
omplete the square to write the equation as y = a(x - h)² + k, where the vertex is (h, k).
求顶点纵坐标时，将 x 值代回方程，或者配方写成 y = a(x - h)² + k，其中顶点为 (h, k)。
Graph sketching should show the shape, intercepts, vertex, and axis of symmetry. A clear sketch can earn method marks even if later algebra slips.
画图时应显示开口方向、截距、顶点和对称轴。清晰的草图即使后续代数出错也可能获得方法分。
8. Solving by Graph and Intersections | 图像法求解与交点
f a curve with a line, usually the x-axis. Draw y = ax² + bx + c and read off where y = 0.
图像法利用曲线与直线（通常是 x 轴）的交点。画出 y = ax² + bx + c，读出 y = 0 的位置。
bx + c, substitute the line into the curve to form a quadratic. The number of intersections equals the number of real solutions.
对于直线 y = mx + k 与抛物线 y = ax² + bx + c 的联立方程，将直线代入曲线得到二次方程。交点个数等于实数解个数。
- If the line cuts the curve twice, there are two solutions.
- If the line touches the curve, there is one repeated solution.
- If the line does not meet the curve, there are no real solutions.
- 若直线与曲线相交两次，则有两个解。
- 若直线与曲线相切，则有一个重解。
- 若直线与曲线不相交，则没有实数解。
minant: the quadratic formed by substitution has Δ > 0, Δ = 0, or Δ < 0 accordingly.
这种图像解释与判别式直接相关：代入后得到的二次方程分别有 Δ > 0、Δ = 0 或 Δ < 0。
9. Word Problems and Modelling | 应用题与建模
Quadratic word problems usually involve area, product of consecutive numbers, projectile motion, or profit. Translate the words into an equation, solve it, and then interpret the answer in context.
二次方程应用题通常涉及面积、连续整数的乘积、抛体运动或利润。将文字转化为方程，求解，再结合情境解释答案。
has length x + 3 and width x. If its area is 40, then x(x + 3) = 40, so x² + 3x - 40 = 0.
常见设定：矩形长为 x + 3、宽为 x。若面积为 40，则 x(x + 3) = 40，所以 x² + 3x - 40 = 0。
x² + 3x - 40 = 0 → (x + 8)(x - 5) = 0 → x = -8 or x = 5
gative, reject x = -8. The valid solution is x = 5, so the width is 5 and the length is 8.
由于长度不能为负，舍去 x = -8。有效解为 x = 5，因此宽为 5，长为 8。
make sense. In projectile problems, t ≥ 0; in area problems, dimensions must be positive.
始终检查单位和两个根是否合理。抛体问题中 t ≥ 0；面积问题中尺寸必须为正。
10. Exam-Style Practice and Common Errors | 考试风格练习与常见错误
Typical exam questions ask students to factorise, solve by completing the square, use the quadratic formula, find the discriminant, sketch a graph, or solve a modelling problem. Marks are awarded for method as well as answer.
典型考题要求学生因式分解、用配方法求解、使用求根公式、求判别式、画图或解建模问题。方法分与答案分并重。
Common errors include: forgetting that a ≠ 0; losing a negative sign when identifying b or c; dividing only part of the numerator by 2a; forgetting to subtract (b ÷ 2)² when completing the square; and using the discriminant without writing the equation in standard form.
常见错误包括：忘记 a ≠ 0；确定 b 或 c 时丢失负号；只将分子的一部分除以 2a；配方时忘记减去 (b ÷ 2)²；以及未写成标准形式就使用判别式。
- Practice 1: Solve x² - 7x + 12 = 0 by factorising.
- Practice 2: Solve 2x² + 5x - 3 = 0 using the formula.
- Practice 3: Find the discriminant of 3x² - 6x + 3 = 0 and state the number of roots.
- Practice 4: Complete the square for x² - 8x + 11 and state the minimum value.
- Practice 5: A number and its square add to 56. Form and solve a quadratic equation.
- 练习 1：用因式分解解 x² - 7x + 12 = 0。
- 练习 2：用公式解 2x² + 5x - 3 = 0。
- 练习 3：求 3x² - 6x + 3 = 0 的判别式，并说明根的个数。
- 练习 4：对 x² - 8x + 11 配方，并写出最小值。
- 练习 5：一个数与它的平方之和为 56。建立并解二次方程。
11. Teaching Sequence and Differentiation | 教学顺序与分层教学
A logical teaching sequence is: expanding brackets, factorising, completing the square, the quadratic formula, the discriminant, graph sketching, and finally modelling. Each step should include short retrieval practice from the previous one.
合理的教学顺序是：展开括号、因式分解、配方法、求根公式、判别式、图像绘制，最后是建模。每一步都应包含对前一步的简短回顾练习。
For students who need support, emphasise factorising and graph reading first, and provide scaffolded templates for the formula. For confident students, extend to proving the formula, deriving the vertex form, and solving inequalities such as x² - 5x + 6 > 0.
对需要支持的学生，先强调因式分解和图像读取，并提供公式填空模板。对有信心 的学生，可拓展到证明公式、推导顶点形式，以及解 x² - 5x + 6 > 0 之类的不等式。
Classroom activities can include matching cards for factorised and expanded forms, mini-whiteboard races for the formula, and graphing software to explore how a, b, and c affect the parabola.
课堂活动可包括因式分解式与展开式的配对卡片、公式的小白板竞速，以及用绘图软件探究 a、b、c 如何影响抛物线。
12. Revision Checklist and Summary | 复习清单与总结
Use this checklist before a test: I can expand and factorise quadratics; I can complete the square and identify the vertex; I can use the quadratic formula accurately; I can calculate and interpret the discriminant; I can sketch a quadratic graph; I can solve a word problem and reject unsuitable roots.
考前使用这份清单：我能展开和因式分解二次式；我能配方并找出顶点；我能准确使用求根公式；我能计算并解释判别式；我能画出二次函数图像；我能解应用题并舍去不合适的根。
The key message of Teacher's Book 600 is that quadratics are not isolated tricks. They are one connected topic: factorising, completing the square, the formula, the discriminant, and the graph are different views of the same mathematical object.
教师用书600 的核心信息是：二次方程不是孤立的技巧。它们是一个相互联系的主题：因式分解、配方法、求根公式、判别式和图像，都是同一个数学对象的不同视角。
Regular practice with mixed questions is the most reliable way to improve. Aim to explain each method in words, not just perform the steps, because IGCSE examiners reward clear communication and correct reasoning.
定期练习混合题型是最可靠的提升方式。要尝试用文字解释每种方法，而不仅仅是执行步骤，因为 IGCSE 考官重视清晰的表达和正确的推理。
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