Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解一元二次方程：因式分解、配方法与求根公式

📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解一元二次方程：因式分解、配方法与求根公式

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and even in word problems involving areas and motion. Mastering three main methods – factorising, completing the square, and using the quadratic formula – gives you a reliable toolkit for any exam question. This article explains each method step by step and shows how to choose the fastest approach under timed conditions.

二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、坐标几何，甚至涉及面积和运动的应用题中。掌握三种主要方法——因式分解、配方法和求根公式——可以为你提供应对任何考试题目的可靠工具箱。本文将逐步解释每种方法，并展示如何在限时考试中选择最快的解题路径。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the unknown x is 2. If a were zero, the equation would become linear. Examples include x² − 5x + 6 = 0 and 2x² + 3x − 4 = 0.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，且 a ≠ 0。未知数 x 的最高次数为 2。如果 a 为零，方程就会变成一次方程。例如 x² − 5x + 6 = 0 和 2x² + 3x − 4 = 0。

ax² + bx + c = 0, a ≠ 0

The solutions of a quadratic equation are called its roots. A quadratic can have two real roots, one repeated real root, or no real roots depending on the value of the discriminant b² − 4ac.

二次方程的解称为它的根。根据判别式 b² − 4ac 的值，二次方程可以有两个实根、一个重根，或没有实根。

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

Before solving, always rearrange the equation into standard form ax² + bx + c = 0. Move every term to one side so that the other side equals zero. This makes the coefficients a, b and c easy to identify.

在求解之前，始终将方程整理成标准形式 ax² + bx + c = 0。把每一项移到一边，使另一边等于零。这样可以方便地确定系数 a、b 和 c。

For example, 3x² − 7 = 2x becomes 3x² − 2x − 7 = 0. Here a = 3, b = −2 and c = −7. Be careful with signs: b is negative because the x term is subtracted, and c is negative because 7 is subtracted on the left side.

例如，3x² − 7 = 2x 可化为 3x² − 2x − 7 = 0。此时 a = 3，b = −2，c = −7。要注意符号：因为 x 项被减去，所以 b 为负；因为左边减去 7，所以 c 为负。

3. Solving by Factorising | 因式分解法

If a quadratic factorises easily, this is usually the fastest method. We look for two numbers that multiply to give ac and add to give b. For simple quadratics where a = 1, we need two numbers that multiply to c and add to b.

如果二次方程容易因式分解，这通常是最快的方法。我们寻找两个数，使它们相乘等于 ac，相加等于 b。对于 a = 1 的简单二次方程，我们需要两个数，使它们相乘等于 c，相加等于 b。

Example: solve x² − 5x + 6 = 0. We need two numbers whose product is +6 and sum is −5. The numbers are −2 and −3, so x² − 5x + 6 = (x − 2)(x − 3) = 0.

例题：解 x² − 5x + 6 = 0。我们需要两个数，使它们的乘积为 +6，和为 −5。这两个数是 −2 和 −3，因此 x² − 5x + 6 = (x − 2)(x − 3) = 0。

By the zero product property, if (x − 2)(x − 3) = 0, then x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

根据零乘积性质，如果 (x − 2)(x − 3) = 0，则 x − 2 = 0 或 x − 3 = 0，因此 x = 2 或 x = 3。

Always check by substituting both roots back into the original equation. Both satisfy x² − 5x + 6 = 0, so the solutions are correct.

始终将两个根代回原方程进行检验。两者都满足 x² − 5x + 6 = 0，所以解是正确的。

4. Factorising When a Is Not 1 | 当 a 不等于 1 时的因式分解

When a ≠ 1, we can still factorise by finding two numbers that multiply to ac and add to b, then splitting the middle term. For example, solve 2x² + 7x + 3 = 0.

当 a ≠ 1 时，我们仍然可以通过寻找两个数（乘积为 ac，和为 b）来分解，然后拆分中间项进行因式分解。例如，解 2x² + 7x + 3 = 0。

Here a = 2, b = 7 and c = 3, so ac = 6. We need two numbers that multiply to 6 and add to 7: they are 6 and 1. Split the middle term: 2x² + 6x + x + 3 = 0.

这里 a = 2，b = 7，c = 3，所以 ac = 6。我们需要两个数，乘积为 6，和为 7：它们是 6 和 1。拆分中间项：2x² + 6x + x + 3 = 0。

Now factorise in pairs: 2x(x + 3) + 1(x + 3) = 0, which gives (2x + 1)(x + 3) = 0. Therefore x = −1/2 or x = −3.

现在成对提取公因式：2x(x + 3) + 1(x + 3

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）