📚 Chapter Review 2: Quadratics | 第二章复习：二次函数
This chapter review covers the essential concepts of quadratic functions, equations, and inequalities from the Edexcel A‑Level Pure Mathematics syllabus. You will revisit factorisation, completing the square, the quadratic formula, the discriminant, sketching parabolas and solving quadratic inequalities. Mastering these tools is fundamental for success in calculus and applied modules.
本章复习涵盖 Edexcel A‑Level 纯数学课程大纲中二次函数、方程和不等式的核心内容。你将重新梳理因式分解、配方法、求根公式、判别式、抛物线草图以及二次不等式的解法。掌握这些工具是学好微积分与应用模块的基础。
1. Standard Form and the Discriminant | 标准形式与判别式
= b² − 4ac determines the nature of the roots of the quadratic equation ax² + bx + c = 0.
二次表达式可以写成标准形式 ax² + bx + c，其中 a ≠ 0。判别式 Δ = b² − 4ac 决定了二次方程 ax² + bx + c = 0 根的性质。
y one real root (a repeated root). When Δ < 0, the equation has no real roots, but two complex conjugate roots.
当 Δ > 0 时，方程有两个不相等的实根。若 Δ = 0，则恰好有一个实根（重根）。当 Δ < 0 时，方程没有实根，但有一对共轭复根。
+ 1 = 0 has Δ = (−4)² − 4(2)(1) = 16 − 8 = 8 > 0, so it possesses two distinct real roots.
例如，方程 2x² − 4x + 1 = 0 的判别式 Δ = (−4)² − 4(2)(1) = 16 − 8 = 8 > 0，因此它有两个不等的实根。
2. Solving by Factorisation | 因式分解法
When a quadratic expression can be written as a product of two linear factors, solving the equation is straightforward. Set each factor equal to zero and solve for x.
当二次表达式可以写成两个一次因式的乘积时，解方程就很简单。令每个因式等于零并解出 x。
stance, x² − 5x + 6 = 0 factorises to (x − 2)(x − 3) = 0, giving solutions x = 2 or x = 3.
例如，x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0，解得 x = 2 或 x = 3。
Always check by expanding the factors to ensure they match the original quadratic. If the coefficient of x² is not 1, factorise out any common factor first or use the ac method.
务必通过展开因式来检验是否与原二次式匹配。如果 x² 的系数不是 1，应先提取公因式或使用 ac 方法。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This form immediately reveals the vertex and the line of symmetry of the parabola.
配方法将二次式改写为 a(x + p)² + q 的形式。这种形式能立即揭示抛物线的顶点和对称轴。
= 9 to obtain (x + 3)² − 4. The vertex is at (−3, −4) and the axis of symmetry is x = −3.
对于 x² + 6x + 5，加上并减去 (6/2)² = 9，得到 (x + 3)² − 4。顶点为 (−3, −4)，对称轴为 x = −3。
x² and x terms first: 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5.
当 a > 1 时，先提取 x² 和 x 项的公因数 a：2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。
4. Quadratic Formula | 求根公式
The solutions to ax² + bx + c = 0 are given by the quadratic formula:
x = (−b ± √b² − 4ac) / (2a)
x = (−b ± √b² − 4ac) / (2a)
This formula works for all quadratics, including those that do not factorise neatly. Always simplify the square root where possible and watch for sign errors when substituting b.
这一公式适用于所有二次方程，包括不能整齐分解的情形。尽可能化简平方根，并注意代入 b 时的符号错误。
For 3x² + 5x − 2 = 0, a = 3, b = 5, c = −2. Substituting gives x = [−5 ± √25 + 24]/6 = [−5 ± √49]/6 = [−5 ± 7]/6, so x = 1/3 or x = −2.
对于 3x² + 5x − 2 = 0，a = 3，b = 5，c = −2。代入得 x = [−5 ± √25 + 24]/6 = [−5 ± √49]/6 = [−5 ± 7]/6，解得 x = 1/3 或 x = −2。
5. Using the Discriminant to Analyse Roots | 用判别式分析根的情况
The discriminant gives insight without solving the equation. If Δ is a perfect square, the roots are rational; otherwise they involve surds.
判别式无需解方程就能提供信息。如果 Δ 是完全平方数，则根为有理数；否则包含根式。
s of k for which a quadratic has equal roots. Setting Δ = 0 yields the required condition.
考试题常会问实根的个数，或求使二次方程有等根的 k 值。令 Δ = 0 便能得出所需条件。
For x² + kx + 9 = 0 to have a repeated root, k² − 36 = 0, so k = ±6.
对于 x² + kx + 9 = 0 有重根，就是 k² − 36 = 0，得 k = ±6。
Also, a positive Δ ensures two points of intersection between a parabola and the x‑axis; a negative Δ tells you the graph lies entirely above or below the axis.
此外，正的 Δ 保证抛物线与 x 轴有两个交点；负的 Δ 则表明图像完全在 x 轴上方或下方。
6. Sketching Quadratic Graphs | 画二次函数图像
y = ax² + bx + c is a parabola. The sign of a determines whether it opens upwards (a > 0, ∪) or downwards (a < 0, ∩).
y = ax² + bx + c 的图像是一条抛物线。a 的符号决定开口向上（a > 0，∪）还是向下（a < 0，∩）。
Key features to label: the y‑ and the vertex. The vertex can be found by completing the square or by using x = −b/(2a).
需要标出的关键特征：y 轴截距 (0, c)、根（如果有）、以及顶点。顶点可以通过配方法或用 x = −b/(2a) 求得。
For y = −x² + 4x − 3, a < 0, so the parabola is ∩‑shaped. The y‑ is −3; the roots are x = 1 and x = 3; the vertex is at (2, 1). Plot these points smoothly.
对于 y = −x² + 4x − 3，a < 0，抛物线呈 ∩ 形。y 轴截距为 −3；根为 x = 1 和 x = 3；顶点在 (2, 1)。平滑地画出这些点。
7. Finding the Vertex and Line of Symmetry | 求顶点与对称轴
From completed‑ ry is x = h. This makes it easy to identify the maximum or minimum value of the quadratic.
由完全平方形式 a(x − h)² + k 可知，顶点为 (h, k)，对称轴为 x = h。这有助于识别二次函数的最大值或最小值。
adratic has a minimum value of k at x = h. If a < 0, it has a maximum value of k at x = h.
若 a > 0，二次函数在 x = h 处取最小值 k；若 a < 0，则在 x = h 处取最大值 k。
To convert f(x) = 2x² − 8x + 5, factor the ‘ ete the square: 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3. Vertex = (2, −3), minimum value = −3.
将 f(x) = 2x² − 8x + 5 转化时，从前两项提取因数 2：2(x² − 4x) + 5。完成平方：2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3。顶点 (2, −3)，最小值 −3。
8. Quadratic Inequalities | 二次不等式
Solving a quadratic inequality such as x² − 4x − 5 > 0 involves finding the roots and testing intervals. First solve the corresponding equation to obtain critical values.
求解如 x² − 4x − 5 > 0 的二次不等式时，需要先求根并检验区间。先解相应的方程，得到临界值。
Factorise: (x − 5)(x + 1) > 0. The roots are −1 and 5. These split the number line into three regions: x < −1, −1 < x < 5, x > 5. Using a sign table or test points gives the solution x < −1 or x > 5.
因式分解得 (x − 5)(x + 1) > 0。根为 −1 和 5。它们将数轴分成三个区间：x < −1、−1 < x < 5、x > 5。使用符号表或检验点可得解为 x < −1 或 x > 5。
e critical values are included in the solution set. Always sketch a quick parabola to confirm the region above or below the axis.
对于含有 ≤ 或 ≥ 的不等式，临界值包含在解集中。务必快速画出抛物线草图，确认位于轴上方还是下方的区域。
9. Modelling with Quadratics | 用二次函数建模
Quadratic models appear in projectile motion, economics, and geometry problems. You often need to formulate a quadratic expression, find its maximum or minimum, and interpret the result.
二次模型常出现在抛体运动、经济学和几何问题中。你通常需要建立二次表达式，求其最大值或最小值，并解释结果。
mpleting the square gives A = −(x − 10)² + 100, so the maximum area is 100 m² when x = 10.
对于周长为 40 米的矩形场地，面积 A = x(20 − x) 得出 A = −x² + 20x。配方得 A = −(x − 10)² + 100，因此当 x = 10 时最大面积为 100 平方米。
Always state your final answer in the context of the problem, including units. Ignore any algebraic solutions that lie outside the feasible domain (e.g., negative lengths).
务必在题目情境中给出最终答案，包括单位。忽略任何超出可行域（例如负长度）的代数解。
10. Common Pitfalls and Exam Tips | 常见错误与应试技巧
Mishandling signs when substituting into the quadratic formula, especially with negative b, is a frequent error. Write brackets around negative values to avoid mistakes.
代入求根公式时符号处理不当，尤其是 b 为负数的情况，是常见错误。在负数周围加括号可避免出错。
When completing the square, remember to adjust the constant term correctly. Many students forget to multiply the added constant by the coefficient a before adding to the constant term.
配方时，记住要正确调整常数项。许多学生忘记先将加上的常数乘以系数 a，然后再与常数项合并。
For inequalities, always check the direction of the inequality when multiplying or dividing by a negative. Represent solutions on a number line and use set notation where required.
解不等式时，当乘以或除以负数，务必检查不等号的方向。用数轴表示解，并在需要时使用集合符号。
11. Summary of Key Techniques | 关键技巧总结
| Technique | Approach |
|---|---|
| Factorisation | Write as product of two binomials; set each to zero. |
| Quadratic Formula | x = [−b ± √b²−4ac]/(2a); always gives roots. |
| Completing the Square | Rewrite as a(x + p)² + q to find vertex and solve. |
| Discriminant | Δ = b²−4ac; number and nature of roots. |
| Inequalities | Find critical values, test intervals, sketch graph. |
These core methods are not isolated; they reinforce each other. Fluency in all of them allows you to choose the most efficient route for any quadratic problem.
这些核心方法并非彼此孤立，而是相互强化。熟练掌握所有方法能让你针对任何二次问题选择最有效率的路径。
12. Practice and Further Revision | 练习与进阶复习
Work through mixed exercises that require switching between forms: standard, factorised, and completed‑square. Pay special attention to questions involving parameters and hidden quadratics (e.g., exponential substitutions).
要练习那些需要在标准式、因式分解式和完全平方式之间转换的混合练习题。特别关注涉及参数和隐藏二次型（如指数代换）的问题。
Regularly revisit the discriminant in unfamiliar contexts, such as proving a quadratic is always positive or finding the range of a rational function. This deepens your understanding and prepares you for the synoptic A‑Level papers.
定期在不熟悉的情境中重温判别式，例如证明二次函数恒正或求有理函数的值域。这能加深理解，为 A‑Level 的综合试卷做好准备。
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