Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

Quadratic equations are a central topic in IGCSE Mathematics. They appear in algebra, geometry, area problems, projectile motion and many real-world contexts. This revision guide explains how to recognise a quadratic equation, how to solve it by factorising, completing the square and the quadratic formula, and how to interpret the discriminant and graph. Worked examples and exam tips are included to help you build confidence.

二次方程是 IGCSE 数学的核心主题之一。它们出现在代数、几何、面积问题、抛体运动以及许多实际情境中。本复习指南将讲解如何识别二次方程，如何用因式分解法、配方法和求根公式求解，以及如何解释判别式和图像。文中包含例题和考试技巧，帮助你建立信心。

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 be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic. The word ‘quadratic’ comes from ‘quadratus’, meaning square, because the highest power of the variable x is x².

二次方程是次数为 2 的多项式方程。它可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。如果 a = 0，方程就变成一次方程，不再是二次方程。’quadratic’ 一词来自拉丁语 ‘quadratus’，意思是平方，因为变量 x 的最高次数是 x²。

For example, x² + 3x + 2 = 0, 2x² − 5x = 0 and x² − 9 = 0 are all quadratic equations. Equations such as x³ + x² = 0 or 5x + 2 = 0 are not quadratic because the highest power of x is not 2.

例如，x² + 3x + 2 = 0、2x² − 5x = 0 和 x² − 9 = 0 都是二次方程。像 x³ + x² = 0 或 5x + 2 = 0 这样的方程不是二次方程，因为 x 的最高次数不是 2。

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

The standard form is ax² + bx + c = 0. The term ax² is called the quadratic term, bx is called the linear term, and c is called the constant term. The coefficient a is the leading coefficient. Before solving, always rearrange the equation so that one side is equal to 0.

标准形式为 ax² + bx + c = 0。ax² 叫二次项，bx 叫一次项，c 叫常数项。系数 a 称为首项系数。在求解之前，一定要把方程整理成一边等于 0 的形式。

For example, x² − 4 = 5x should be rearranged as x² − 5x − 4 = 0 before applying any solving method. In this form, a = 1, b = −5 and c = −4.

例如，x² − 4 = 5x 在应用任何求解方法之前，应先整理为 x² − 5x − 4 = 0。在这个形式中，a = 1，b = −5，c = −4。

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3. Solving by Factorising | 因式分解法求解

When a quadratic expression can be written as a product of two linear factors, we can solve it quickly. First write the equation in standard form, then factorise the quadratic expression into two brackets, and finally set each bracket equal to zero.

当二次式可以写成两个一次因式的乘积时，就能快速求解。首先把方程写成标准形式，然后把二次式分解成两个括号的乘积，最后令每个括号等于零。

For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0. Then set each factor equal to zero: x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。然后令每个因式等于零：x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。

A harder example is 2x² + 7x + 3 = 0. We look for two numbers that multiply to 2 × 3 = 6 and add to 7: the numbers are 6 and 1. Split the middle term: 2x² + 6x + x + 3 = 0, then factorise by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. So x = −1/2 or x = −3.

一个较难的例子是 2x² + 7x + 3 = 0。我们需要两个数，乘积为 2 × 3 = 6，和为 7：这两个数是 6 和 1。把中间项拆开：2x² + 6x + x + 3 = 0，然后分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。所以 x = −1/2 或 x = −3。

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4. The Zero Product Property | 零乘积性质

The factorising method relies on the zero product property: if m × n = 0, then m = 0 or n = 0, or both. This is only true because the product is zero. If the product equals any other number, we cannot simply set each factor equal to that number.

因式分解法依赖零乘积性质：如果 m × n = 0，那么 m = 0 或 n = 0，或两者都为零。只有当乘积为零时才能这样用。如果乘积等于其他数，就不能简单地令每个因式等于那个数。

For instance, if (x − 2)(x + 1) = 0, then x − 2 = 0 or x + 1 = 0, so x = 2 or x = −1. But if (x − 2)(x + 1) = 4, we cannot say x − 2 = 4 or x + 1 = 4; the left side must first be expanded and rearranged.

例如，如果 (x − 2)(x + 1) = 0，那么 x − 2 = 0 或 x + 1 = 0，所以 x = 2 或 x = −1。但如果 (x − 2)(x + 1) = 4，就不能说 x − 2 = 4 或 x + 1 = 4；必须先把左边展开并整理。

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5. When Factorising is Not Easy | 当因式分解不容易时

Some quadratics do not factorise into simple integer factors, for example x² − 2x − 1 = 0. In such cases, we can use completing the square or the quadratic formula. We can also use the discriminant to check whether a quadratic has rational roots before trying to factorise.

有些二次方程不能分解为简单的整数因式，例如 x² − 2x − 1 = 0。这时可以使用配方法或求根公式。我们也可以在尝试因式分解之前，先用判别式判断二次方程是否有有理数根。

If the discriminant b² − 4ac is a perfect square, the quadratic factorises nicely with rational factors. If it is not a perfect square but still positive, the roots are irrational, so factorising with integers will not work.

如果判别式 b² − 4ac 是一个完全平方数，二次式就能很好地分解为有理因式。如果它不是完全平方数但仍是正数，根就是无理数，因此无法用整数因式分解。

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6. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method is especially useful when the quadratic does not factorise easily, and it is also needed to find the turning point of a parabola.

配方法把 ax² + bx + c 写成 a(x + p)² + q 的形式。当二次方程不容易因式分解时，这种方法特别有用；同时，求抛物线的顶点也需要用到配方法。

For a simple case, x² + 6x + 4 = 0 can be written as (x + 3)² − 9 + 4 = 0, so (x + 3)² = 5. Taking square roots gives x + 3 = ±√5, so x = −3 ± √5.

举个简单例子，x² + 6x + 4 = 0 可以写成 (x + 3)² − 9 + 4 = 0，所以 (x + 3)² = 5。开平方得到 x + 3 = ±√5，因此 x = −3 ± √5。

For a coefficient a ≠ 1, first divide or factor out the coefficient of x². Example: 2x² + 8x + 5 = 0 becomes 2(x² + 4x) + 5 = 0. Complete the square inside the bracket: x² + 4x = (x + 2)² − 4, so 2[(x + 2)² − 4] + 5 = 0, which gives 2(x + 2)² − 3 = 0, so x = −2 ± √(3/2).

当首项系数 a ≠ 1 时，先把 x² 的系数提出来。例如：2x² + 8x + 5 = 0 变为 2(x² + 4x) + 5 = 0。在括号内完成配方：x² + 4x = (x + 2)² − 4，所以 2[(x + 2)² − 4] + 5 = 0，得到 2(x + 2)² − 3 = 0

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