IGCSE Mathematics: Quadratic Equations — Complete Guide | IGCSE 数学：二次方程完全指南

IGCSE Mathematics: Quadratic Equations — Complete Guide

A quadratic equation is one of the most fundamental topics in IGCSE Mathematics. It appears across both the Core and Extended syllabi and is essential for success in topics ranging from algebra to kinematics. This guide covers everything you need to know: standard form, solving methods, the discriminant, graphing, and exam-style applications.

二次方程是 IGCSE 数学中最基础的主题之一。它同时出现在 Core 和 Extended 教学大纲中，对于从代数到运动学等各个主题的成功至关重要。本指南涵盖了您需要了解的所有内容：标准形式、求解方法、判别式、图形绘制以及考试风格的应用题。

1. What is a Quadratic Equation? 什么是二次方程？

A quadratic equation is a polynomial equation of degree 2. Its standard form is:

二次方程是一个二次多项式方程。其标准形式为：

ax^2 + bx + c = 0

where a, b, and c are constants, and a != 0. If a = 0, the equation becomes linear, not quadratic.

其中 a、b 和 c 是常数，且 a != 0。如果 a = 0，方程就变成了线性方程，而不是二次方程。

In IGCSE Mathematics, you will encounter quadratics with integer coefficients, fractional coefficients, and sometimes with parameters. The variable x represents the unknown that we solve for.

在 IGCSE 数学中，您会遇到具有整数系数、分数系数，有时还带有参数的二次方程。变量 x 代表我们要求解的未知数。

2. Three Methods for Solving Quadratic Equations 求解二次方程的三种方法

The Cambridge IGCSE syllabus requires you to master three distinct methods for solving quadratic equations. Each method has its strengths, and exam questions often specify which method to use.

剑桥 IGCSE 教学大纲要求您掌握三种不同的二次方程求解方法。每种方法都有其优势，考试题目通常会指定使用哪种方法。

2.1 Factorisation 因式分解法

Factorisation is the most elegant method and works when the quadratic can be expressed as a product of two linear factors.

因式分解是最优雅的方法，当二次方程可以表示为两个线性因式的乘积时适用。

Example 示例： Solve x^2 – 5x + 6 = 0

Step 1: Find two numbers that multiply to give +6 and add to give -5.
These are -2 and -3.

步骤 1：找到两个数，乘积为 +6，和为 -5。
这两个数是 -2 和 -3。

Step 2: Rewrite as (x – 2)(x – 3) = 0
Step 3: Set each factor to zero: x – 2 = 0 or x – 3 = 0
Step 4: x = 2 or x = 3

步骤 2：重写为 (x – 2)(x – 3) = 0
步骤 3：将每个因式设为零：x – 2 = 0 或 x – 3 = 0
步骤 4：x = 2 或 x = 3

Key exam tip: Always check your answers by substituting them back into the original equation. 2^2 – 5(2) + 6 = 4 – 10 + 6 = 0 (correct).

考试关键提示：始终将答案代入原方程进行验证。2^2 – 5(2) + 6 = 4 – 10 + 6 = 0（正确）。

2.2 Completing the Square 配方法

Completing the square is a powerful algebraic technique that rewrites the quadratic in the form a(x + p)^2 + q = 0. This method is particularly useful for finding the vertex of a parabola and for deriving the quadratic formula.

配方法是一种强大的代数技巧，它将二次方程重写为 a(x + p)^2 + q = 0 的形式。此方法对于求抛物线的顶点和推导二次公式特别有用。

Example 示例： Solve x^2 + 6x + 2 = 0 by completing the square.

Step 1: Move the constant to the right: x^2 + 6x = -2
Step 2: Take half the coefficient of x (6/2 = 3) and square it (3^2 = 9). Add to both sides:
x^2 + 6x + 9 = -2 + 9
(x + 3)^2 = 7

步骤 1：将常数移到右边：x^2 + 6x = -2
步骤 2：取 x 系数的一半（6/2 = 3）并平方（3^2 = 9）。两边同时加上：
x^2 + 6x + 9 = -2 + 9
(x + 3)^2 = 7

Step 3: Take the square root of both sides: x + 3 = +/-sqrt(7)
Step 4: x = -3 +/- sqrt(7)

步骤 3：两边开平方：x + 3 = +/-sqrt(7)
步骤 4：x = -3 +/- sqrt(7)

The answers in exact (surd) form are x = -3 + sqrt(7) and x = -3 – sqrt(7). The IGCSE Extended syllabus expects answers to be left in surd form unless otherwise instructed.

精确（根式）形式的答案为 x = -3 + sqrt(7) 和 x = -3 – sqrt(7)。除非另有说明，IGCSE Extended 教学大纲要求答案保留根式形式。

2.3 The Quadratic Formula 二次公式法

The quadratic formula is the universal method that works for all quadratic equations, regardless of whether they factorise nicely. It is derived from completing the square on the general form ax^2 + bx + c = 0.

二次公式是通用方法，适用于所有二次方程，无论它们是否能很好地因式分解。它是由一般形式 ax^2 + bx + c = 0 通过配方法推导出来的。

x = [-b +/- sqrt(b^2 – 4ac)] / (2a)

This formula is provided on the IGCSE formula sheet, but you must know how to apply it correctly. Let’s work through an example.

该公式在 IGCSE 公式表中提供，但您必须知道如何正确使用它。让我们通过一个例子来学习。

Example 示例： Solve 2x^2 – 4x – 3 = 0

Identify: a = 2, b = -4, c = -3

Substitute into the formula:

x = [-(-4) +/- sqrt((-4)^2 – 4(2)(-3))] / (2 * 2)
x = [4 +/- sqrt(16 + 24)] / 4
x = [4 +/- sqrt(40)] / 4
x = [4 +/- 2*sqrt(10)] / 4
x = 1 +/- sqrt(10)/2

代入公式：
x = [-(-4) +/- sqrt((-4)^2 – 4(2)(-3))] / (2 * 2)
x = [4 +/- sqrt(16 + 24)] / 4
x = [4 +/- sqrt(40)] / 4
x = [4 +/- 2*sqrt(10)] / 4
x = 1 +/- sqrt(10)/2

3. The Discriminant 判别式

The expression under the square root in the quadratic formula, b^2 – 4ac, is called the discriminant. It tells us the nature of the roots without actually solving the equation.

二次公式中平方根下的表达式 b^2 – 4ac 被称为判别式。它告诉我们根的性质，而无需实际求解方程。

Discriminant Value 判别式值	Nature of Roots 根的性质
b^2 – 4ac > 0	Two distinct real roots 两个不同的实根
b^2 – 4ac = 0	One repeated real root 一个重复的实根
b^2 – 4ac < 0	No real roots (two complex roots) 无实根（两个复根）

Exam question pattern 考试题型： “Find the value(s) of k for which the equation has equal roots.” This means set b^2 – 4ac = 0 and solve for k.

“求使方程具有相等根的 k 值。”这意味着令 b^2 – 4ac = 0 并求解 k。

Example: For x^2 + kx + 9 = 0 to have equal roots:
b^2 – 4ac = k^2 – 4(1)(9) = k^2 – 36 = 0
k^2 = 36, so k = +/-6

示例：要使 x^2 + kx + 9 = 0 具有相等根：
b^2 – 4ac = k^2 – 4(1)(9) = k^2 – 36 = 0
k^2 = 36，所以 k = +/-6

4. Graphs of Quadratic Functions 二次函数图像

The graph of y = ax^2 + bx + c is a parabola. Understanding its shape and key features is essential for the IGCSE examination.

y = ax^2 + bx + c 的图像是一条抛物线。理解其形状和关键特征对于 IGCSE 考试至关重要。

4.1 Shape: The Role of ‘a’ 形状：’a’ 的作用

If a > 0, the parabola opens upward (U-shaped, a “happy” parabola) — it has a minimum point.
如果 a > 0，抛物线开口向上（U 形，”快乐”抛物线）— 它有一个最小值点。
If a < 0, the parabola opens downward (inverted-U-shaped, a “sad” parabola) — it has a maximum point.
如果 a < 0，抛物线开口向下（倒U 形，”悲伤”抛物线）— 它有一个最大值点。

4.2 Key Features 关键特征

Y-intercept 与 y 轴的交点： Set x = 0, giving y = c. The point is (0, c).
令 x = 0，得到 y = c。交点为 (0, c)。

X-intercepts (roots) 与 x 轴的交点（根）： Set y = 0 and solve ax^2 + bx + c = 0. These are the solutions to the quadratic equation.
令 y = 0 并求解 ax^2 + bx + c = 0。这些就是二次方程的解。

Vertex (turning point) 顶点（转折点）： The x-coordinate is x = -b/(2a). Substitute this back into the equation to find the y-coordinate. This is the minimum (if a > 0) or maximum (if a < 0) value of the function.
x 坐标为 x = -b/(2a)。将其代回方程求 y 坐标。这是函数的最小值（若 a > 0）或最大值（若 a < 0）。

Axis of symmetry 对称轴： The vertical line x = -b/(2a) that passes through the vertex. The parabola is symmetric about this line.
穿过顶点的垂直线 x = -b/(2a)。抛物线关于此线对称。

5. Solving Quadratic Inequalities 解二次不等式

Quadratic inequalities appear frequently in the IGCSE Extended syllabus. The approach combines solving the related quadratic equation with understanding the shape of the parabola.

二次不等式经常出现在 IGCSE Extended 教学大纲中。解题方法结合了求解相关二次方程和理解抛物线的形状。

Example 示例： Solve x^2 – x – 6 > 0

Step 1: Solve the related equation: x^2 – x – 6 = 0
(x – 3)(x + 2) = 0, so x = 3 or x = -2

步骤 1：求解相关方程：x^2 – x – 6 = 0
(x – 3)(x + 2) = 0，所以 x = 3 或 x = -2

Step 2: Sketch the parabola. Since a = 1 > 0, it opens upward. The roots are at x = -2 and x = 3.

步骤 2：画出抛物线的草图。由于 a = 1 > 0，开口向上。根在 x = -2 和 x = 3 处。

Step 3: The parabola is above the x-axis (y > 0) when x < -2 or x > 3.
Answer: x < -2 or x > 3

步骤 3：当 x < -2 或 x > 3 时，抛物线在 x 轴上方（y > 0）。
答案：x < -2 或 x > 3

Critical exam tip: Many students lose marks by writing “-2 < x < 3” when the question asks for > 0. Always check: if a > 0, the U-shape means the parabola is above the x-axis outside the interval between the roots, and below the x-axis inside the interval.

关键考试提示：许多学生因在题目要求 > 0 时写成 “-2 < x < 3” 而丢分。始终检查：若 a > 0，U 形意味着抛物线在两根之间的区间外于 x 轴上方，在区间内于 x 轴下方。

6. Word Problems Involving Quadratics 涉及二次方程的应用题

IGCSE exam papers frequently feature contextual problems that lead to quadratic equations. These test your ability to translate real-world scenarios into mathematical models.

IGCSE 考试卷经常出现导致二次方程的情境题。这些题目测试您将现实场景转化为数学模型的能力。

6.1 Area Problems 面积问题

Example: A rectangular garden has length 3 m more than its width. Its area is 40 m^2. Find the dimensions.

示例：一个矩形花园的长比宽多 3 米。其面积为 40 平方米。求其尺寸。

Let width = x, then length = x + 3.
Area: x(x + 3) = 40
x^2 + 3x – 40 = 0
(x + 8)(x – 5) = 0
x = 5 (reject x = -8 as length cannot be negative)
Width = 5 m, Length = 8 m

设宽度 = x，则长度 = x + 3。
面积：x(x + 3) = 40
x^2 + 3x – 40 = 0
(x + 8)(x – 5) = 0
x = 5（舍去 x = -8，因为长度不能为负）
宽度 = 5 m，长度 = 8 m

6.2 Projectile Motion 抛体运动

Example: A ball is thrown upward from a height of 2 m with an initial velocity of 20 m/s. Its height h (in metres) after t seconds is given by h = -5t^2 + 20t + 2. Find when the ball hits the ground.

示例：一个球从 2 米高处向上抛出，初速度为 20 m/s。其高度 h（以米为单位）在 t 秒后由 h = -5t^2 + 20t + 2 给出。求球何时落地。

When the ball hits the ground, h = 0:
-5t^2 + 20t + 2 = 0
5t^2 – 20t – 2 = 0 (multiply by -1 for convenience)

Using the formula: t = [20 +/- sqrt(400 + 40)] / 10 = [20 +/- sqrt(440)] / 10
t is approximately [20 +/- 20.98] / 10
t is approximately 4.10 seconds (reject negative time)

当球落地时，h = 0：
-5t^2 + 20t + 2 = 0
5t^2 – 20t – 2 = 0（为了简便，乘以 -1）
使用公式：t = [20 +/- sqrt(400 + 40)] / 10 = [20 +/- sqrt(440)] / 10
t 约等于 [20 +/- 20.98] / 10
t 约等于 4.10 秒（舍去负时间）

7. Common Mistakes to Avoid 常见错误避免

Forgetting a != 0: If the coefficient of x^2 is zero, the equation is linear, not quadratic. Always check before applying quadratic methods.
忘记 a != 0：如果 x^2 的系数为零，则方程为线性方程而非二次方程。在应用二次方程方法前务必检查。
Incorrect factorisation: When expanding (x + p)(x + q), the constant term is p * q and the coefficient of x is p + q. Students often mix these up.
因式分解错误：展开 (x + p)(x + q) 时，常数项为 p * q，x 的系数为 p + q。学生经常混淆这两者。
Losing the +/- sign: Remember that sqrt(x^2) = |x|, so when you solve (x + 3)^2 = 7, you get x + 3 = +/-sqrt(7), not just +sqrt(7).
丢失 +/- 号：记住 sqrt(x^2) = |x|，因此当您解 (x + 3)^2 = 7 时，得到 x + 3 = +/-sqrt(7)，而不仅仅是 +sqrt(7)。
Not rejecting invalid solutions: In context problems, negative answers for lengths, times, or other physical quantities must be rejected.
未舍去无效解：在情境题中，长度、时间或其他物理量的负值答案必须被舍去。
Arithmetic errors in the formula: Be especially careful with the “-4ac” part. If c is negative, -4ac becomes positive. Double-check your signs.
公式中的算术错误：对 “-4ac” 部分要特别小心。如果 c 为负，-4ac 变为正。仔细检查您的符号。

8. Past Paper Practice 历年真题练习

The best way to prepare for IGCSE quadratic equation questions is through consistent practice with past papers. Here are some typical question styles you should master:

准备 IGCSE 二次方程题目的最佳方法是通过历年真题进行持续练习。以下是您应该掌握的几种典型题型：

Type 1 — Factorisation (Core & Extended):
Solve x^2 – 7x + 10 = 0
题型 1 — 因式分解（Core 和 Extended）：
求解 x^2 – 7x + 10 = 0

Type 2 — Quadratic Formula (Extended):
Solve 3x^2 + 5x – 1 = 0, giving answers correct to 2 decimal places.
题型 2 — 二次公式（Extended）：
求解 3x^2 + 5x – 1 = 0，答案精确到小数点后两位。

Type 3 — Completing the Square (Extended):
Express x^2 + 8x – 5 in the form (x + p)^2 + q and hence find the minimum value.
题型 3 — 配方法（Extended）：
将 x^2 + 8x – 5 表示为 (x + p)^2 + q 的形式，并求出最小值。

Type 4 — Discriminant (Extended):
Find the values of k for which the equation 2x^2 + kx + 8 = 0 has no real roots.
题型 4 — 判别式（Extended）：
求使方程 2x^2 + kx + 8 = 0 无实根的 k 值。

Summary 总结

Quadratic equations are a cornerstone of IGCSE Mathematics. By mastering the three solution methods — factorisation, completing the square, and the quadratic formula — and understanding the discriminant and graph features, you will be well-prepared for any question the exam may present. Regular practice with past papers is the key to building speed and accuracy.

二次方程是 IGCSE 数学的基石。通过掌握三种求解方法——因式分解、配方法和二次公式——并理解判别式和图像特征，您将为考试中可能出现的任何问题做好充分准备。通过历年真题的持续练习是提高速度和准确性的关键。

This article covers content from the Cambridge IGCSE Mathematics (0580/0980) syllabus. For more IGCSE study resources, browse our collection of past papers and revision guides at aleveler.com.

本文涵盖了剑桥 IGCSE 数学（0580/0980）教学大纲的内容。如需更多 IGCSE 学习资源，请浏览我们在 aleveler.com 上的历年真题和复习指南合集。

IGCSE Mathematics: Quadratic Equations — Complete Guide | IGCSE 数学：二次方程完全指南