Quadratic Equations: Solving, Graphing, and Exam Skills | IGCSE 数学：二次方程求解、图像与应试技巧

📚 Quadratic Equations: Solving, Graphing, and Exam Skills | IGCSE 数学：二次方程求解、图像与应试技巧

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to area problems and projectile motion. This article explains how to recognise a quadratic, solve it by factorising, completing the square, and using the quadratic formula, and also shows how to connect algebra with graphs and real-world questions.

二次方程贯穿 IGCSE 数学大纲，从代数与图像到面积问题和抛体运动都有涉及。本文将讲解如何识别二次方程，如何用因式分解、配方法和求根公式求解，并展示如何把代数与图像以及现实情境题联系起来。

1. Recognising a Quadratic Equation | 认识二次方程

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2, and this is what makes the equation quadratic rather than linear.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。x 的最高次数为 2，这正是该方程为二次方程而非一次方程的原因。

For example, 2x² − 5x + 3 = 0 is quadratic because the term 2x² has degree 2. Equations such as x² = 9 or 3x − x² = 4 can be rearranged into standard form before solving.

例如，2x² − 5x + 3 = 0 是二次方程，因为 2x² 的次数为 2。像 x² = 9 或 3x − x² = 4 这样的方程可以先整理为标准形式再求解。

Always check that a is not zero. If a = 0, the equation becomes bx + c = 0, which is linear and has only one solution.

一定要检查 a 不等于 0。如果 a = 0，方程就变成 bx + c = 0，这是一次方程并且只有一个解。

2. Solving by Factorising | 因式分解法求解

Factorising is often the fastest method when the quadratic has simple integer roots. The idea is to write ax² + bx + c as a product of two brackets, then set each bracket equal to 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。其思路是把 ax² + bx + c 写成两个括号的乘积，然后令每个括号等于零。

For x² − 5x + 6 = 0, we look for two numbers that multiply to 6 and add to −5. These are −2 and −3, so the factorised form is (x − 2)(x − 3) = 0.

对于 x² − 5x + 6 = 0，我们要找两个数，使它们的乘积为 6，和为 −5。这两个数是 −2 和 −3，因此因式分解形式为 (x − 2)(x − 3) = 0。

When the coefficient of x² is not 1, such as 2x² + 7x + 3, multiply a and c to get 6. We need factors of 6 that add to 7: 6 and 1. Then split the middle term and factor by grouping: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

当 x² 的系数不是 1 时，例如 2x² + 7x + 3，先把 a 和 c 相乘得 6。我们需要找到两个数，乘积为 6 且和为 7：即 6 和 1。然后拆分中间项并分组分解：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

After factorising, set each bracket to zero to find x = −1/2 or x = −3.

因式分解后，令每个括号为零，得到 x = −1/2 或 x = −3。

3. The Zero Product Property | 零乘积性质

The key rule behind factorising is the zero product property: if two expressions multiply to give zero, then at least one of them must be zero. This only works when the other side of the equation is exactly 0.

因式分解背后的核心规则是零乘积性质：如果两个式子的乘积为零，那么其中至少有一个必须为零。这只有在方程另一侧恰好为 0 时才成立。

From (x − 2)(x − 3) = 0, we write x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

由 (x − 2)(x − 3) = 0，我们得到 x − 2 = 0 或 x − 3 = 0，从而 x = 2 或 x = 3。

If the equation is not equal to zero, for example (x − 1)(x + 2) = 4, you cannot simply set each bracket equal to 4. You must expand, rearrange to standard form, and then factorise: x² + x − 2 = 4 becomes x² + x − 6 = 0, which factorises to (x + 3)(x − 2) = 0.

如果方程不等于零，例如 (x − 1)(x + 2) = 4，你不能简单地把每个括号设为 4。必须展开、移项为标准形式，再因式分解：x² + x − 2 = 4 变为 x² + x − 6 = 0，分解为 (x + 3)(x − 2) = 0。

4. Completing the Square | 配方法

Completing the square is useful for solving quadratics that do not factorise nicely, and it is essential for finding the turning point of a quadratic graph. The goal is to rewrite x² + bx + c in the form (x + p)² + q.

配方法适用于不易因式分解的二次方程，而且在求二次图像顶点时必不可少。目标是把 x² + bx + c 改写为 (x + p)² + q 的形式。

For x² + 6x + 2 = 0, take half of the coefficient of x, which is 3, square it to get 9, and add and subtract it: x² + 6x + 9 − 9 + 2 = (x + 3)² − 7 = 0.

对于 x² + 6x + 2 = 0，取 x 系数的一半即 3，平方得 9，再加 9 减 9：x² + 6x + 9 − 9 + 2 = (x + 3)² − 7 = 0。

Solving (x + 3)² =

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