Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与公式法

📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与公式法

Quadratic equations are among the most important algebraic skills in IGCSE Maths. They appear in pure algebra problems, coordinate geometry, quadratic graphs and real-life applications such as area optimisation or projectile motion. This article explains three core methods – factorising, completing the square and the quadratic formula – and shows you how to choose the best one under timed exam conditions.

二次方程是 IGCSE 数学最重要的代数技能之一。它们出现在纯代数题、坐标几何、二次函数图像以及实际应用中，例如面积优化或抛体运动。本文讲解三种核心方法：因式分解法、配方法和公式法，并帮助你在限时考试中选择最佳方法。

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

A quadratic equation is any equation in which the highest power of the unknown, usually x, is 2. It can be written in the general form shown below, where a, b and c are constants and a does not equal zero.

二次方程是指未知数（通常是 x）的最高次数为 2 的方程。它可以写成下面的一般形式，其中 a、b 和 c 是常数，且 a 不等于 0。

ax² + bx + c = 0

If a = 0, the equation becomes linear, such as bx + c = 0, and is no longer quadratic. For example, 2x² + 3x − 5 = 0 is quadratic, while 3x − 5 = 0 is linear.

如果 a = 0，方程就变成一次方程，例如 bx + c = 0，不再是二次方程。例如，2x² + 3x − 5 = 0 是二次方程，而 3x − 5 = 0 是一次方程。

2. Writing the Equation in Standard Form | 将方程整理为标准形式

Before solving, you should rearrange all terms onto one side so the other side equals zero. The standard form is ax² + bx + c = 0 with the terms in descending powers of x.

解题前，应先将所有项移到一边，使另一边等于 0。标准形式为 ax² + bx + c = 0，各项按 x 的降幂排列。

Example: 3x + x² = 7 should be rearranged as x² + 3x − 7 = 0.

例如：3x + x² = 7 应整理为 x²

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