Solving Quadratic Equations by Factorising and the Quadratic Formula | 二次方程的因式分解与求根公式求解

📚 Solving Quadratic Equations by Factorising and the Quadratic Formula | 二次方程的因式分解与求根公式求解

In IGCSE Mathematics, quadratic equations are a key part of the Algebra syllabus. They appear in direct solving questions, word problems involving area, and questions about the graph of a parabola. A quadratic equation is any equation that can be rearranged into the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. Mastering two main methods – factorising and the quadratic formula – gives you a reliable toolkit for the exam.

在 IGCSE 数学中，二次方程是代数教学大纲的核心部分。它们出现在直接求解题、涉及面积的应用题以及抛物线图像题中。二次方程是任何可以整理成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数且 a ≠ 0。掌握因式分解和求根公式这两种主要方法，可以为你提供可靠的考试工具。

1. Recognising Quadratic Equations | 识别二次方程

A quadratic equation has a highest power of x equal to 2. The standard form is written as shown below. If the coefficient of x² is zero, the equation becomes linear, not quadratic.

二次方程中 x 的最高次数为 2。其标准形式如下所示。如果 x² 的系数为零，方程就变成一次方程，而不是二次方程。

ax² + bx + c = 0, where a ≠ 0

In IGCSE questions, you may need to expand brackets or collect like terms before writing the equation in standard form. For example, 2(x² − 3) = 4x can be rearranged to 2x² − 4x − 6 = 0.

在 IGCSE 题目中，你可能需要先展开括号或合并同类项，再把方程写成标准形式。例如，2(x² − 3) = 4x 可以整理为 2x² − 4x − 6 = 0。

2. The Zero Product Principle | 零乘积原理

The factorising method depends on the zero product principle. If two numbers or expressions multiply to give zero, then at least one of them must be zero. This allows us to split a factorised quadratic into two simple linear equations.

因式分解法依赖于零乘积原理。如果两个数或两个式子的乘积为零，那么其中至少有一个必须为零。这使我们能够把因式分解后的二次方程拆成两个简单的一次方程。

If p × q = 0, then p = 0 or q = 0

In an exam, always set the equation equal to zero before factorising. For example, x² − 3x + 2 = 0 is ready, but x² − 3x = −2 needs rearranging first.

在考试中，因式分解前一定要先把方程化为等于零的形式。例如，x² − 3x + 2 = 0 已经准备好，但 x² − 3x = −2 需要先移项。

3. Factorising x² + bx + c | 因式分解 x² + bx + c

When the coefficient of x² is 1, the quadratic looks like x² + bx + c. To factorise, look for two numbers m and n such that m × n = c and m + n = b. Then x² + bx + c = (x + m)(x + n).

当 x² 的系数为 1 时，二次式形如 x² + bx + c。因式分解时，要找两个数 m 和 n，使得 m × n = c 且 m + n = b。于是 x² + bx + c = (x + m)(x + n)。

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