Quadratic Functions & Equations: Complete IGCSE Guide | 二次函数与二次方程：IGCSE 完整指南

📚 Quadratic Functions & Equations: Complete IGCSE Guide | 二次函数与二次方程：IGCSE 完整指南

Quadratic functions and equations form a cornerstone of the IGCSE Mathematics syllabus. This guide provides a comprehensive, exam-focused walkthrough of everything you need to know — from factorisation and the quadratic formula to graphing and word problems.

二次函数与二次方程是 IGCSE 数学考纲的核心内容。本指南将带你系统梳理所有必考知识点——从因式分解、求根公式到图像绘制与应用题，一网打尽。

1. What Is a Quadratic Expression? | 什么是二次表达式？

A quadratic expression is a polynomial of degree 2, meaning the highest power of the variable is 2. Its general form is written as:

二次表达式是次数为 2 的多项式，即变量的最高次幂为 2。它的一般形式写作：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are constants, and a must not be zero — if a = 0, the expression becomes linear. For example, 3x² + 5x − 2 = 0 is quadratic, while 2x + 7 = 0 is not.

其中 a、b、c 为常数，且 a 不能为零——若 a = 0，则表达式变为一次式。例如，3x² + 5x − 2 = 0 是二次方程，而 2x + 7 = 0 则不是。

2. Standard Form vs. Vertex Form | 标准形式与顶点形式

Quadratic functions can be expressed in two key forms. The standard form y = ax² + bx + c is useful for identifying the y-intercept and applying the quadratic formula. The vertex form y = a(x − h)² + k directly reveals the turning point of the parabola.

二次函数有两种重要形式。标准形式 y = ax² + bx + c 便于确定 y 截距并应用求根公式；顶点形式 y = a(x − h)² + k 则能直接显示出抛物线的顶点。

Form / 形式	Equation / 方程	Key Feature / 关键特征
Standard 标准	y = ax² + bx + c	y-intercept at (0, c)
Vertex 顶点	y = a(x − h)² + k	Vertex at (h, k)

The vertex form is especially powerful when sketching graphs: if a > 0, the parabola opens upward with a minimum at (h, k); if a < 0, it opens downward with a maximum at (h, k).

顶点形式在画图时尤为强大：若 a > 0，抛物线开口向上，在 (h, k) 处取得最小值；若 a < 0，抛物线开口向下，在 (h, k) 处取得最大值。

3. Solving by Factorisation | 因式分解法求解

Factorisation is often the fastest method when the quadratic has simple integer roots. The goal is to rewrite ax² + bx + c as the product of two brackets, then set each bracket to zero.

当二次方程存在简单的整数根时，因式分解往往是最快的方法。目标是将 ax² + bx + c 改写为两个括号相乘的形式，然后令每个括号分别等于零。

Example / 示例: Solve x² − 5x + 6 = 0 by factorisation.

We look for two numbers that multiply to give +6 and add to give −5. These numbers are −2 and −3. Therefore:

我们寻找两个数，它们相乘为 +6，相加为 −5。这两个数是 −2 和 −3。因此：

(x − 2)(x − 3) = 0

Setting each factor to zero gives x = 2 or x = 3. Always expand your brackets to check the answer.

令每个因式等于零，得到 x = 2 或 x = 3。务必展开括号验证答案是否正确。

4. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula provides a universal solution method. For any quadratic equation ax² + bx + c = 0, the solutions are given by:

当因式分解困难或无法进行时，求根公式提供了一种通用的解法。对于任意二次方程 ax² + bx + c = 0，其解由以下公式给出：

x = (−b ± √(b² − 4ac)) / 2a

Example / 示例: Solve 2x² − 4x − 3 = 0 using the quadratic formula.

Here a = 2, b = −4, c = −3. Substituting into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

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

Simplifying: x = (4 ± 2√10) / 4 = 1 ± √10 / 2. This gives two irrational roots.

化简得：x = (4 ± 2√10) / 4 = 1 ± √10 / 2，得到两个无理数根。

5. Completing the Square | 配方法

Completing the square transforms a quadratic into vertex form, making it easy to identify maximum and minimum points. The process begins by halving the coefficient of x.

配方法将二次式转化为顶点形式，从而轻松确定最大值和最小值。该过程首先需要将 x 的系数除以 2。

Example / 示例: Write x² + 6x + 5 in the form (x + p)² + q.

Half of 6 is 3, so we write (x + 3)². Since (x + 3)² = x² + 6x + 9, we subtract 9 to compensate:

6 的一半是 3，因此我们写出 (x + 3)²。由于 (x + 3)² = x² + 6x + 9，减去 9 以作补偿：

x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4

This tells us the vertex is at (−3, −4), and the minimum value of the function is −4.

由此可知顶点坐标为 (−3, −4)，函数的最小值为 −4。

6. The Discriminant | 判别式

The discriminant, denoted by Δ = b² − 4ac, determines the number and nature of roots without solving the equation. It reveals whether the roots are real, equal, or unreal.

判别式，记作 Δ = b² − 4ac，无需解方程即可判断根的个数与性质。它能揭示根是实数、相等还是虚数。

If Δ > 0: the equation has two distinct real roots.

若 Δ > 0：方程有两个不同的实数根。
If Δ = 0: the equation has exactly one repeated real root.

若 Δ = 0：方程有一个重根（两个相等的实数根）。
If Δ < 0: the equation has no real roots (two complex roots).

若 Δ < 0：方程没有实数根（有两个复数根）。

Example / 示例: Determine the nature of roots for 3x² − 2x + 5 = 0.

Here Δ = (−2)² − 4 × 3 × 5 = 4 − 60 = −56 < 0, so the equation has no real roots.

此处 Δ = (−2)² − 4 × 3 × 5 = 4 − 60 = −56 < 0，因此方程没有实数根。

7. Sketching Quadratic Graphs | 绘制二次函数图像

To sketch a quadratic graph accurately, you need to find the key features: the roots, the y-intercept, and the vertex. Follow these steps:

要准确绘制二次函数图像，需要找出关键特征点：根、y 截距和顶点。请按以下步骤操作：

Find the y-intercept by substituting x = 0.

代入 x = 0 求 y 截距。
Find the roots by setting y = 0, using factorisation or the quadratic formula.

令 y = 0，通过因式分解或求根公式求根。
Find the vertex using x = −b / 2a, then substitute back to find y.

用 x = −b / 2a 求顶点的 x 坐标，再代回原式求 y 坐标。
Plot the points and draw a smooth curve.

标出这些点并绘制平滑曲线。

Vertex x-coordinate: x = −b / 2a

Remember: when a > 0 the parabola has a U-shape; when a < 0 it has an inverted U-shape.

注意：当 a > 0 时抛物线呈 U 形；当 a < 0 时呈倒 U 形。

8. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities extends the concept of solving equations. After finding the roots, you use a number line or the graph to determine the range of x that satisfies the inequality.

解二次不等式是方程求解的延伸。在找到根之后，借助数轴或函数图像来确定满足不等式的 x 取值范围。

Example / 示例: Solve x² − x − 6 > 0.

First, factorise: (x − 3)(x + 2) > 0. The roots are x = 3 and x = −2. Sketching the parabola shows it lies above the x-axis when x < −2 or x > 3.

首先因式分解：(x − 3)(x + 2) > 0。根为 x = 3 和 x = −2。绘制抛物线可知：当 x < −2 或 x > 3 时，函数图像在 x 轴上方。

Solution: x < −2 or x > 3

If the inequality is ≥ 0, we include the roots: x ≤ −2 or x ≥ 3.

若不等式为 ≥ 0，则需包含根：x ≤ −2 或 x ≥ 3。

9. Applications in Word Problems | 应用题中的二次函数

Quadratic equations appear frequently in real-world contexts, from projectile motion to area optimisation. The key is to translate the problem into a quadratic equation and then solve it using the most appropriate method.

二次方程在现实场景中频繁出现，从抛体运动到面积优化问题。关键在于将实际问题转化为二次方程，然后使用最合适的方法求解。

Example / 示例: A rectangle has an area of 40 m². Its length is 3 m longer than its width. Find the dimensions.

示例：一个长方形的面积为 40 平方米。其长比宽多 3 米。求其尺寸。

Let the width be x. Then the length is (x + 3), and the area equation is:

设宽为 x，则长为 (x + 3)，面积方程为：

x(x + 3) = 40 → x² + 3x − 40 = 0

Factorising gives (x + 8)(x − 5) = 0, so x = 5 (rejecting x = −8, as length cannot be negative). The rectangle is 5 m by 8 m.

因式分解得 (x + 8)(x − 5) = 0，所以 x = 5（舍去 x = −8，因为长度不能为负）。该长方形宽 5 米、长 8 米。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Avoiding common pitfalls is essential for maximising your marks. Here are the most frequent errors students make:

避免常见陷阱对于最大化得分至关重要。以下是学生最常犯的错误：

Forgetting to set the equation to zero before factorising — this is the single most common mistake.

因式分解前忘记将方程化为零——这是最常见的错误。
Losing negative signs when substituting into the quadratic formula — always use brackets carefully.

代入求根公式时丢失负号——务必小心使用括号。
Misidentifying the sign of a when determining the parabola’s direction.

在判断抛物线开口方向时弄错 a 的符号。
When using the formula x = −b / 2a, forgetting to substitute back to find the y-coordinate of the vertex.

使用公式 x = −b / 2a 求得 x 坐标后，忘记代回求顶点 y 坐标。
In word problems, failing to reject negative solutions that have no physical meaning.

在应用题中，未舍去没有物理意义的负数解。

Always check your solutions by substituting them back into the original equation. In GDC-supported exams, use your graphic calculator to verify the graph matches your algebraic answer.

始终将解代回原方程进行验证。在允许使用图形计算器的考试中，可用计算器绘制图像来确认代数答案与图形一致。

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Quadratic Functions & Equations: Complete IGCSE Guide | 二次函数与二次方程：IGCSE 完整指南