IGCSE Maths: Quadratic Equations Masterclass | IGCSE 数学：二次方程精讲

📚 IGCSE Maths: Quadratic Equations Masterclass | IGCSE 数学：二次方程精讲

Quadratic equations are one of the highest-impact topics in IGCSE Mathematics. They link algebra, graphing, and real-world problem solving, and exam questions often ask you to solve, analyse, or model quadratics. Mastering the main solution methods will give you confidence across both Core and Extended papers.

二次方程是 IGCSE 数学中分值高、综合性强的重要主题。它将代数、图像与现实问题建模连接在一起，考试中经常要求你求解、分析或建立二次方程模型。掌握主要解法能帮助你在 Core 和 Extended 试卷中都更有把握。

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

A quadratic equation is a polynomial equation of degree 2. It can always be written as ax² + bx + c = 0, where x is the unknown and a, b, c are constants. The defining feature is the x² term, which produces a parabola when the equation is graphed.

二次方程是最高次数为 2 的多项式方程。它总可以写成 ax² + bx + c = 0 的形式，其中 x 是未知数，a、b、c 是常数。其核心特征是含有 x² 项，因此它的图像是一条抛物线。

If a = 0, the x² term disappears and the equation becomes linear. For example, 3x² – 2x + 1 = 0 is quadratic, but 5x + 2 = 0 is linear and x³ + x² = 0 is cubic, not quadratic.

如果 a = 0，x² 项消失，方程就退化为一次方程。例如，3x² – 2x + 1 = 0 是二次方程，而 5x + 2 = 0 是一次方程，x³ + x² = 0 是三次方程，不是二次方程。

2. Standard Form and Key Terms | 标准形式与关键术语

The standard form is:

标准形式为：

ax² + bx + c = 0, a ≠ 0

Here, a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. Before solving, you should rearrange the equation so that one side equals zero and the terms are in descending powers of x.

其中 a 是二次项系数，b 是一次项系数，c 是常数项。求解前，应先将方程整理为一边等于 0，且各项按 x 的降幂排列。

a controls the shape and direction of the parabola — a 控制抛物线的形状和开口方向。
b affects the position of the vertex — b 影响顶点的位置。
c is the y-intercept of the graph — c 是图像的 y 轴截距。

For example, in 2x² – 4x + 1 = 0, we have a = 2, b = -4 and c = 1. Show these values clearly before using any formula.

例如，在 2x² – 4x + 1 = 0 中，a = 2，b = -4，c = 1。在使用任何公式前，请清楚地写出这些数值。

3. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic has simple integer roots. Write the quadratic as a product of two linear brackets, then set each bracket equal to zero. This works because if a product equals zero, at least one factor must be zero.

当二次方程有简单的整数根时，因式分解通常是最快的方法。先将二次式写成两个一次因式的乘积，再令每个因式等于零。这是基于一个原理：若乘积为零，则至少有一个因式为零。

Example 1:

Solve x² + 5x + 6 = 0

Factorise the left side:

(x + 2)(x + 3) = 0

Set each factor equal to zero:

x + 2 = 0 → x = -2

x + 3 = 0 → x = -3

Example 1 中文：解 x² + 5x + 6 = 0。将左边因式分解为 (x + 2)(x + 3) = 0。令每个因式等于零，得到 x = -2 或 x = -3。

Example 2 with a leading coefficient greater than 1:

Solve 2x² + 7x + 3 = 0

Factorise:

(2x + 1)(x + 3) = 0

Set each factor to zero:

2x + 1 = 0 → x = -1/2

x + 3 = 0 → x = -3

Example 2 中文：解 2x² + 7x + 3 = 0。因式分解得到 (2x + 1)(x + 3) = 0。令每个因式等于零，解得 x = -1/2 或 x = -3。

4. Solving by Completing the Square | 配方法

Completing the square transforms ax² + bx + c into the form a(x + p)² + q. This method is especially useful when factorising is difficult, when you need to find the vertex of a parabola, or when a question asks for answers in surd form.

配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式。当因式分解困难、需要求抛物线顶点，或题目要求用根式形式给出答案时，配方法特别有用。

Example:

Solve x² + 6x + 1 = 0

Move the constant to the right:

x² + 6x = -1

Add (6/2)² = 9 to both sides:

x² + 6x + 9 = 8

Write as a perfect square:

(x + 3)² = 8

Take the square root and solve:

x + 3 = ±√8

x = -3 ± √8 = -3 ± 2√2

中文示例：解 x² + 6x + 1 = 0。先把常数移到右边得到 x² + 6x = -1。两边加上 (6/2)² = 9，得到 x² + 6x + 9 = 8。左边写成完全平方 (x + 3)² = 8。开方并求解，得到 x = -3 ± √8 = -3 ± 2√2。

5. The Quadratic Formula | 求根公式

When factorising is not obvious, use the quadratic formula. It works for every quadratic equation and is often the safest method on the non-calculator paper when the roots are irrational or messy.

当因式分解不明显时，使用求根公式。它适用于所有二次方程，而且在根是无理数或比较复杂的非计算器试卷中，通常是最稳妥的方法。

x = (-b ± √(b² – 4ac)) / 2a

Substitute a, b and c carefully, including negative signs. Then simplify the square root and the fraction. Always write the formula before substituting to earn method marks.

代入 a、b、c 时要特别注意负号，然后化简根号和分式。答题时先写出公式再代入，以便获得方法分。

Example:

Solve 2x² – 3x – 5 = 0

Here a = 2, b = -3, c = -5. Substitute:

x = (-(-3) ± √((-3)² – 4 × 2 × (-5))) / (2 × 2)

x = (3 ± √(9 + 40)) / 4 = (3 ± √49) / 4 = (3 ± 7) / 4

This gives two answers:

x = (3 + 7) / 4 = 10/4 = 5/2

x = (3 – 7) / 4 = -4/4 = -1

中文示例：解 2x² – 3x – 5 = 0。这里 a = 2，b = -3，c = -5。代入求根公式得到 x = (3 ± √(9 + 40)) / 4 = (3 ± 7) / 4，因此 x = 5/2 或 x = -1。

6. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant D = b² – 4ac tells you how many real roots a quadratic equation has. This is important for questions that ask you to determine the nature of the roots without solving the equation fully.

判别式 D = b² – 4ac 可以判断二次方程实数根的个数。有些题目不要求完全求解，只要求判断根的性质，这时判别式就非常重要。

Discriminant	Nature of roots	根的性质
D > 0	Two distinct real roots	两个不相等的实数根
D = 0	One repeated real root	一个重根（两个相等实根）
D < 0	No real roots	没有实数根

Example: For x² – 6x + 9 = 0, D = (-6)² – 4 × 1 × 9 = 36 – 36 = 0, so the equation has a repeated root x = 3. For x² + x + 1 = 0, D = 1 – 4 = -3, so it has no real roots.

示例：对于 x² – 6x + 9 = 0，D = (-6)² – 4 × 1 × 9 = 36 – 36 = 0，因此方程有重根 x = 3。对于 x² + x + 1 = 0，D = 1 – 4 = -3，因此没有实数根。

7. Word Problems Leading to Quadratics | 应用题建模

Many IGCSE exam questions ask you to form a quadratic from a practical situation. The key steps are: define the unknown, write an equation from the given information, solve it, and then check whether the answers make sense in the real context.

许多 IGCSE 考题要求你从实际问题中建立二次方程。关键步骤是：设未知数，根据已知信息列出方程，求解，然后检验答案在实际情境中是否合理。

Example: The length of a rectangle is 3 cm longer than its width, and the area is 40 cm². Let the width be x cm, so the length is x + 3 cm. The area equation is:

示例：一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x cm，则长为 x + 3 cm。面积方程为：

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IGCSE Maths: Quadratic Equations Masterclass | IGCSE 数学：二次方程精讲