Mastering Quadratic Equations and Functions | 二次方程与函数全面解析

📚 Mastering Quadratic Equations and Functions | 二次方程与函数全面解析

Quadratic equations and functions form one of the most heavily tested topics in IGCSE Mathematics. From factorisation to the quadratic formula, from sketching parabolas to solving inequalities, this single topic connects algebra, graphs and geometry. In this revision guide, we break down every key skill you need, with clear bilingual explanations and worked examples.

二次方程与函数是 IGCSE 数学中考查频率最高的专题之一。从因式分解到求根公式，从抛物线作图到解不等式，这一个专题将代数、图像与几何紧密联系在一起。在本复习指南中，我们将逐一拆解所有关键考点，配以清晰的双语讲解与例题。

1. Understanding Quadratic Expressions | 认识二次表达式

A quadratic expression is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which gives the expression its name (‘quad’ refers to ‘square’).

二次表达式是指形如 ax² + bx + c 的代数式，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此称为”二次”（quad 意为”平方”）。

The standard form of a quadratic equation is set equal to zero:

ax² + bx + c = 0, where a ≠ 0

For example, in the equation 2x² – 5x + 3 = 0, we have a = 2, b = -5 and c = 3. Recognising these coefficients is essential before you attempt any solution method.

例如，在方程 2x² – 5x + 3 = 0 中，a = 2，b = -5，c = 3。在尝试任何解法之前，必须先准确识别这些系数。

2. Solving by Factorisation | 因式分解法求解

Factorisation is the fastest method when the roots are integers or simple fractions. The idea is to rewrite ax² + bx + c as a product of two binomials, then use the zero product property.

当根为整数或简单分数时，因式分解是最快捷的方法。其核心思想是将 ax² + bx + c 改写为两个二项式的乘积，然后利用”零乘积性质”求解。

For monic quadratics (where a = 1), find two numbers that multiply to give c and add to give b. For non-monic quadratics (a ≠ 1), use factorisation by grouping or trial and error.

对于首项系数为 1 的二次式（a = 1），寻找两个数，使它们的乘积等于 c，和等于 b。对于首项系数不为 1 的二次式（a ≠ 1），可使用分组分解法或试错法。

Worked example: Solve x² – 7x + 12 = 0.

例题：解方程 x² – 7x + 12 = 0。

We need two numbers whose product is 12 and sum is -7. Those numbers are -3 and -4:

我们需要找到两个数，使它们的乘积为 12，和为 -7。这两个数是 -3 和 -4：

(x – 3)(x – 4) = 0 → x = 3 or x = 4

Always check your factorisation by expanding the brackets back out — it takes ten seconds and earns you guaranteed marks.

务必通过展开括号来验算因式分解是否正确——这只需十秒钟，却能确保你稳拿分数。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This form directly reveals the turning point of the graph and is essential for solving equations that do not factorise neatly.

配方法将二次式改写为 a(x + p)² + q 的形式。这一形式能直接揭示图像的顶点坐标，也是解那些无法整洁因式分解的方程的关键工具。

For x² + bx, add and subtract (b/2)²:

对于 x² + bx，加上并减去 (b/2)²：

x² + bx = (x + b/2)² – (b/2)²

Worked example: Solve x² + 6x – 7 = 0 by completing the square.

例题：用配方法解方程 x² + 6x – 7 = 0。

First, rewrite x² + 6x as (x + 3)² – 9. Then the equation becomes (x + 3)² – 9 – 7 = 0, so (x + 3)² = 16. Taking square roots gives x + 3 = ±4, hence x = 1 or x = -7.

首先，将 x² + 6x 改写为 (x + 3)² – 9。于是方程变为 (x + 3)² – 9 – 7 = 0，即 (x + 3)² = 16。两边开平方得 x + 3 = ±4，因此 x = 1 或 x = -7。

Remember: when you take the square root

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