Mastering Quadratics for Edexcel IGCSE Maths | 攻克二次方程与二次函数

📚 Mastering Quadratics for Edexcel IGCSE Maths | 攻克二次方程与二次函数

Quadratics appear in nearly every Edexcel IGCSE Mathematics paper, whether as pure algebra questions, graph-sketching problems, or word problems. Understanding how to expand, factorise, solve and sketch quadratic expressions is therefore one of the most valuable skills you can build. This revision guide breaks the topic into clear steps with the exact notation you need for the exam.

二次函数与二次方程几乎是 Edexcel IGCSE 数学每份试卷的”必考点”，无论是纯代数题、图像题还是应用题都会涉及。掌握展开、因式分解、求解和绘制二次函数图像的方法，是备考中最值得投入的环节。本复习指南将带你分步攻克这一核心专题，并严格使用考试所需的规范符号。

1. Expanding Brackets | 展开括号

To expand two linear brackets, multiply each term in the first bracket by every term in the second bracket. The memory aid FOIL (First, Outer, Inner, Last) helps you avoid missing terms. For example, (x + 4)(x − 3) = x² − 3x + 4x − 12 = x² + x − 12. Always combine the like terms after expanding.

展开两个一次因式时，需要用第一个括号中的每一项去乘第二个括号中的每一项。口诀 FOIL（首项、外项、内项、末项）可以帮助你避免遗漏。例如，(x + 4)(x − 3) = x² − 3x + 4x − 12 = x² + x − 12。展开后一定要合并同类项。

When a bracket is squared, such as (2x + 3)², write it out as (2x + 3)(2x + 3) before expanding. A very common error is to write (2x + 3)² = 4x² + 9, which is wrong. Correctly: 4x² + 12x + 9.

当遇到括号整体平方时，例如 (2x + 3)²，应先写成 (2x + 3)(2x + 3) 再展开。一个常见错误是直接写成 (2x + 3)² = 4x² + 9，这是不对的。正确结果是 4x² + 12x + 9。

2. Factorising Quadratics | 因式分解二次三项式

For a quadratic of the form x² + bx + c, look for two numbers p and q such that p + q = b and pq = c. Then the factorised form is (x + p)(x + q). For instance, factorise x² + 5x + 6: we need two numbers that sum to 5 and multiply to 6, which are 2 and 3, giving (x + 2)(x + 3).

对于形如 x² + bx + c 的二次三项式，需要找到两个数 p 和 q，使得 p + q = b 且 pq = c。分解结果就是 (x + p)(x + q)。例如，分解 x² + 5x + 6：需要找到和为 5、积为 6 的两个数，即 2 和 3，所以结果为 (x + 2)(x + 3)。

If the coefficient of x² is not 1, for example 2x² + 7x + 3, use the method of drawing a 2×2 grid, or the “product and sum” trick: multiply a and c to get 2 × 3 = 6, then find two numbers that multiply to 6 and add to 7 — those are 6 and 1. Split the middle term: 2x² + 6x + x + 3, then factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

如果 x² 的系数不是 1，例如 2x² + 7x + 3，可以使用 2×2 网格法，或者”乘积与和”技巧：先把 a 和 c 相乘，得到 2 × 3 = 6，然后找出两个数，它们相乘为 6 且相加为 7，即 6 和 1。拆分中间项：2x² + 6x + x + 3，再分组提取公因式：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

3. Solving by Factorisation | 因式分解法解方程

To solve ax² + bx + c = 0 by factorisation, first factorise the left-hand side fully. Then apply the zero-product property: if the product of two expressions is zero, at least one of them must be zero. So (x − 3)(x − 4) = 0 gives x − 3 = 0 or x − 4 = 0, hence x = 3 or x = 4.

用因式分解法解 ax² + bx + c = 0，首先要将左侧完全因式分解，然后应用”零乘积性质”：如果两个表达式的乘积为零，则至少有一个为零。因此 (x − 3)(x − 4) = 0 可得 x − 3 = 0 或 x − 4 = 0，从而 x = 3 或 x = 4。

Be careful when the quadratic is set equal to a value other than zero. For example, to solve x² − 5x = 6, rearrange to x² − 5x − 6 = 0 first. Factorising gives (x − 6)(x + 1) = 0, so x = 6 or x = −1. Never factorise before moving all terms to one side.

特别注意：当方程右边不是 0 时，需要先移项。例如，解 x² − 5x = 6，应先将方程整理为 x² − 5x − 6 = 0。因式分解得 (x − 6)(x + 1) = 0，所以 x = 6 或 x = −1。切记不要把右侧的数直接忽略，必须先移项再分解。

4. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula. This method works for every quadratic, even when factorisation is difficult or impossible.

对于任意一元二次方程 ax² + bx + c = 0，解可由求根公式给出。这种方法适用于一切二次方程，即使因式分解困难甚至无法分解时也有效。

x = (−b ± √(b² − 4ac)) / 2a

Worked example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute into the formula:

例题：解方程 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入求根公式得：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2

Leave the answer in surd form unless the question asks for a decimal. On the Edexcel IGCSE exam, exact answers using surds and π are always credited.

除非题目要求保留小数，否则答案应保留根号（无理数）形式。在 Edexcel IGCSE 考试中，使用根号和 π 的精确答案通常都会获得满分。

5. Completing the Square | 配方法

Completing the square rewrites x² + bx + c in the form (x + b/2)² − (b/2)² + c. For example, x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. This form is extremely useful because it immediately reveals the vertex of the graph and the minimum or maximum value.

配方法将 x² + bx + c 改写为 (x + b/2)² − (b/2)² + c 的形式。例如，x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。这种形式极为重要，因为它能直接揭示图像的顶点以及函数的最小值或最大值。

When the coefficient of x² is not 1, first factor it out from the x² and x terms only. For 2x² − 8x + 5, factor out 2: 2(x² − 4x) + 5, then complete the square inside the bracket: 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3. The vertex is therefore (2, −3) and the minimum value is −3.

当 x² 的系数不为 1 时，先只将 x² 项和 x 项提取公因子。对于 2x² − 8x + 5，先提出 2：2(x² − 4x) + 5，再对括号内配方：2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3。因此顶点为 (2, −3)，最小值为 −3。

6. The Discriminant | 判别式 Δ

The expression inside the square root of the quadratic formula is called the discriminant, written as Δ = b² − 4ac. Its value tells you how many real roots a quadratic equation has, without solving it fully.

求根公式中根号内的部分称为判别式，记作 Δ = b² − 4ac。它的取值可以告诉你二次方程有多少个实数根，而无需完整求解。

If Δ > 0, there are two distinct real roots — the graph cuts the x-axis at two points.
If Δ = 0, there is exactly one repeated root — the graph touches the x-axis at the vertex.
If Δ < 0, there are no real roots — the graph never crosses the x-axis.

当 Δ > 0 时，方程有两个不同的实数根，图像与 x 轴有两个交点。
当 Δ = 0 时，方程有一个重根，图像在顶点处与 x 轴相切。
当 Δ < 0 时，方程没有实数根，图像与 x 轴无交点。

This idea is often tested with the phrase “the equation has no real roots” or “the line is a tangent”. For instance, if x² + kx + 9 = 0 has a repeated root, then Δ = k² − 36 = 0, so k = ±6.

这一知识点常以”方程无实数根”或”直线为切线”等语句来考查。例如，若 x² + kx + 9 = 0 有重根，则 Δ = k² − 36 = 0，解得 k = ±6。

7. Sketching Quadratic Graphs | 二次函数图像的绘制

A quadratic graph y = ax² + bx + c is a parabola. If a > 0 it opens upwards with a minimum point; if a < 0 it opens downwards with a maximum point. For a clear sketch you should find three features: the y-intercept, the x-intercepts (roots), and the vertex.

二次函数 y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时开口向上，有最小值点；当 a < 0 时开口向下，有最大值点。画草图时需要确定三个关键要素：y 截距、x 截距（根）以及顶点。

y-intercept: set x = 0; the intercept is (0, c).
x-intercepts: set y = 0 and solve ax² + bx + c = 0 by factorising or using the formula.
Vertex: find using completing the square, or use x = −b / 2a and substitute back to find y.

y 截距：令 x = 0，截距为 (0, c)。
x 截距：令 y = 0，通过因式分解或求根公式解 ax² + bx + c = 0。
顶点：用配方法求得，或利用 x = −b / 2a 求出 x 坐标后再代回求 y。

Example: sketch y = x² − 2x − 3. Factorise to get y = (x − 3)(x + 1), so the roots are x = 3 and x = −1. The y-intercept is (0, −3). The vertex lies midway between the roots at x = 1; substituting gives y = 1 − 2 − 3 = −4. So the vertex is (1, −4).

例如：绘制 y = x² − 2x − 3 的草图。因式分解得 y = (x − 3)(x + 1)，所以根为 x = 3 和 x = −1。y 截距为 (0, −3)。顶点位于两个根的中点 x = 1 处，代入得 y = 1 − 2 − 3 = −4。因此顶点为 (1, −4)。

8. Solving Word Problems | 应用题求解

Many exam problems translate a real-world situation into a quadratic equation. The strategy is to define the unknown variable, write down the equation from the given relationships, then solve it and interpret the answer in the context of the problem.

许多考试题目需要将实际情境转化为二次方程。解题策略是：先设未知数，再根据题意中的等量关系列出方程，然后求解，最后结合实际背景解释答案。

Example: The length of a rectangle is 4 cm longer than its width, and its area is 45 cm². Let the width be w. Then length = w + 4, so w(w + 4) = 45. Expanding gives w² + 4w − 45 = 0. Factorising: (w + 9)(w − 5) = 0, so w = −9 or w = 5. Since length cannot be negative, w = 5 and the length is 9 cm.

例题：一个长方形的长比宽长 4 cm，面积为 45 cm²。设宽为 w，则长为 w + 4，因此 w(w + 4) = 45。展开得 w² + 4w − 45 = 0。因式分解：(w + 9)(w − 5) = 0，所以 w = −9 或 w = 5。由于长度不能为负数，取 w = 5，长为 9 cm。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

The most frequent errors in quadratic questions are sign errors, forgetting to rearrange equations to zero, losing solutions when dividing by a variable, and misremembering the quadratic formula. Here are the tips that save the most marks.

二次函数题目中最常见的错误包括：符号错误、忘记将方程移项为 0、在方程两边除以含变量的项导致漏根，以及记错求根公式。下面这些技巧能在考场上帮你保住宝贵的分数。

Always check your factorisation by expanding it back; this takes 10 seconds and catches most errors.
When solving a quadratic, expect two answers unless the discriminant is zero; if you only find one root, check your work.
Never divide both sides by x or by any expression containing x — you will lose one root. Factorise instead.
In graph-sketching questions, label the intercepts and vertex with coordinates; Edexcel mark schemes reward clear labelling.
Use the simplest method: factorise if possible, otherwise use the formula; completing the square is best when the vertex is required.

分解完成后立即重新展开验证，这只需 10 秒，却能发现绝大多数错误。
解二次方程时应预期得到两个答案（除非判别式为 0）；如果只求出一个根，请回头检查。
切勿在方程两边同除以 x 或任何含 x 的表达式，否则会漏掉一个根。应当改用因式分解。
在画图像题中，务必标注截距和顶点的坐标，Edexcel 评分标准对清晰的标注会给分。
优先选择最简单的方法：能分解就分解，否则用求根公式；当题目要求顶点时，配方法最为合适。

Finally, practise past paper questions on quadratics until the methods become automatic. The more familiar you are with each form — factorised, completed-square, and standard — the faster and more accurately you will answer on exam day.

最后，坚持练习历年真题中的二次函数题，直到各种方法熟练到近乎本能。你对因式分解形式、配方形式、标准形式三种形态越熟悉，考场上就能答得越快、越准。

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Mastering Quadratics for Edexcel IGCSE Maths | 攻克二次方程与二次函数