Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most frequently tested topics in IGCSE Mathematics. Mastering the different methods of solving quadratic equations is essential for both Paper 2 and Paper 4, and it forms the foundation for many higher-level topics such as functions, inequalities, and calculus.

二次方程是 IGCSE 数学中考查频率最高的专题之一。掌握解二次方程的多种方法，对 Paper 2 和 Paper 4 都至关重要，同时也为函数、不等式、微积分等更高阶内容打下坚实基础。

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 written as:

二次方程是次数为 2 的多项式方程，其标准形式写作：

ax² + bx + c = 0, where a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 is crucial because if a = 0, the equation becomes linear, not quadratic.

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 非常关键，因为若 a = 0，方程就变成了线性方程，而非二次方程。

For example, 2x² + 3x − 5 = 0 is quadratic, while 2x + 3 = 0 is not.

例如，2x² + 3x − 5 = 0 是二次方程，而 2x + 3 = 0 不是。

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

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0. This means moving all terms to one side and simplifying. Only then can factorisation or the quadratic formula be applied correctly.

在求解之前，务必先将方程整理成标准形式 ax² + bx + c = 0，即将所有项移到等号一侧并化简。只有这样才能正确使用因式分解或求根公式。

• The solutions of a quadratic equation are called the roots or solutions.

  二次方程的解称为方程的根（roots）或解（solutions）。

• The expression b² − 4ac is called the discriminant, denoted by Δ (delta).

  表达式 b² − 4ac 称为判别式，用 Δ（德尔塔）表示。

• A quadratic equation always has at most two roots.

  二次方程最多有两个根。

When a quadratic equation is given in non-standard form, such as 2x² = 5x − 3, you must first rewrite it as 2x² − 5x + 3 = 0 before applying any solution method.

当二次方程以非标准形式给出时，例如 2x² = 5x − 3，必须先改写为 2x² − 5x + 3 = 0，再使用任何解法。

3. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic has simple integer roots. The idea is to express ax² + bx + c as a product of two linear expressions.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。其核心思想是将 ax² + bx + c 表示为两个一次表达式的乘积。

Consider the example x² − 5x + 6 = 0. We look for two numbers that multiply to 6 and add to −5. These numbers are −2 and −3, so we write:

以 x² − 5x + 6 = 0 为例。我们需要找到两个数，它们相乘得 6，相加得 −5。这两个数是 −2 和 −3，于是写出：

(x − 2)(x − 3) = 0

Since the product is zero, at least one factor must be zero. Therefore, x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

由于乘积为零，至少有一个因子必须为零。因此 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。

For quadratics where a ≠ 1, such as 2x² + 7x + 3 = 0, factorisation requires more care:

对于 a ≠ 1 的二次方程，如 2x² + 7x + 3 = 0，因式分解需要更细心：

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

Hence x = −1/2 or x

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