Solving Quadratic Equations: A Complete IGCSE Guide | 解二次方程：IGCSE 完整指南

📚 Solving Quadratic Equations: A Complete IGCSE Guide | 解二次方程：IGCSE 完整指南

Quadratic equations are a fundamental topic in IGCSE Mathematics. This guide, adapted from the G-5 teacher’s handbook (page 1099), provides a structured approach to solving these equations. It covers factorisation, completing the square, the quadratic formula, the discriminant, and practical applications. Each concept is paired with clear examples to support both teaching and independent learning.

二次方程是 IGCSE 数学中的基础重点。本指南改编自 G-5 教师手册（第1099页），提供了解这类方程的结构化方法。内容涵盖因式分解、配方法、二次公式、判别式以及实际应用。每个概念都配有清晰的例题，以支持教师教学与学生自学。

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

A quadratic equation is a polynomial equation of degree 2, written in the general form:

ax² + bx + c = 0

where a, b and c are real numbers, and a ≠ 0. The value x is the unknown. The highest power of x is 2, which gives the equation its name.

其中 a、b、c 是实数，且 a ≠ 0。x 是未知数，x 的最高次幂是 2，方程因此得名。

• Example: x² – 9 = 0. 示例：x² – 9 = 0。
• Example: 2x² + 3x – 5 = 0. 示例：2x² + 3x – 5 = 0。
• Example: -x² + 4x = 2 (requires rearrangement). 示例：-x² + 4x = 2（需要移项变形）。

Not every equation with x² is quadratic in the strict IGCSE sense; it must be reducible to the form above with no other higher powers.

并非所有含 x² 的方程都是严格意义下的二次方程；必须能化简为上述形式，且不包含更高次幂。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when simple integer factors exist. The steps are simple:

因式分解法在存在简单整数因子时往往是最快的方法。步骤如下：

1. Rewrite the equation in the form ax² + bx + c = 0. 将方程化为 ax² + bx + c = 0 的形式。
2. Factorise the left-hand side. 对左边进行因式分解。
3. Set each factor equal to zero. 令每个因式等于零。
4. Solve the resulting linear equations. 解所得的线性方程。

Example: Solve x² – 5x + 6 = 0.

示例：解 x² – 5x + 6 = 0。

(x – 2)(x – 3) = 0

x = 2 or x = 3

For a leading coefficient other than 1, use the ‘cross multiplication’ method. Example: 2x² + 7x + 3 = 0.

当首项系数不为 1 时，可使用十字相乘法。示例：2x² + 7x + 3 = 0。

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

x = -½ or x = -3

Always check your factors by expanding them mentally before solving.

在求解之前，务必通过展开来检验你的因式是否正确。

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3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic expression as a perfect square plus a constant. This method is useful when the equation cannot be factorised easily.

配方法将一个二次表达式改写为“完全平方加常数”的形式。当方程不易因式分解时，此方法非常有用。

Example: Solve x² + 6x – 7 = 0.

示例：解 x² + 6x – 7 = 0。

First, isolate the x terms and complete the square:

首先，分离含 x 的项并配方：

(x + 3)² – 9 – 7 = 0

(x + 3)² = 16

x + 3 = ±4

x = 1 or x = -7

If the coefficient of x² is not 1, first factor it out. For example: 2x² + 8x – 10 = 0 can be written as 2[(x + 2)² – 9] = 0, leading to x = 1 or x = -5.

如果 x² 的系数不为 1，先将其提出来。例如：2x² + 8x – 10 = 0 可写成 2[(x + 2)² – 9] = 0，解得 x = 1 或 x = -5。

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4. The Quadratic Formula | 二次求根公式

The quadratic formula provides a universal solution for any quadratic equation. For ax² + bx + c = 0, the solutions are:

二次公式为任意二次方程提供了通用解法。对于 ax² + bx + c = 0，解为：

x = [-b ± √(b² – 4ac)] / (2a)

This formula works for all cases, including irrational and complex roots (if allowed by the syllabus). Ensure you substitute a, b, c carefully with signs.

该公式适用于所有情况，包括无理根和（如果大纲允许的）复数根。代入 a、b、c 时务必注意符号。

Example: Solve x² + 2x – 8 = 0 using the formula.

示例：用公式解 x² + 2x – 8 = 0。

x = [-2 ± √(2² – 4 × 1 × (-8))] / (2 × 1)

x = [-2 ± √36] / 2

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