Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程求解

📚 Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程求解

A quadratic equation is one of the central algebra skills tested in IGCSE Mathematics. You will meet it as a standalone solving question or as part of a larger problem on area, motion, or sequences. Mastering the three main techniques – factorising, completing the square, and using the quadratic formula – gives you a reliable toolkit for any question style.

二次方程是 IGCSE 数学代数部分的核心技能之一。它经常作为独立的求解题出现，也会结合面积、运动或数列等问题一起考查。掌握因式分解法、配方法和求根公式这三种主要方法，能让你应对任何题型。

1. What Makes an Equation Quadratic? | 什么使方程成为二次方程

A quadratic equation is an equation of degree 2, which means the highest power of the variable is 2. The most common variable is x, but any letter can be used. The equation must include an x² term; it may also include an x term and a constant term. If the x² term has coefficient zero, the equation is not quadratic.

二次方程是次数为 2 的方程，也就是说未知数的最高次数是 2。最常用的未知数是 x，但也可以使用其他字母。方程中必须含有 x² 项，也可以同时含有 x 项和常数项。如果 x² 项的系数为 0，它就不再是二次方程。

ax² + bx + c = 0, where a ≠ 0

The condition a ≠ 0 is essential. For example, 3x² − 2x + 1 = 0 is quadratic, while 0x² + 4x − 7 = 0 is just the linear equation 4x − 7 = 0.

条件 a ≠ 0 非常重要。例如，3x² − 2x + 1 = 0 是二次方程，而 0x² + 4x − 7 = 0 只是线性方程 4x − 7 = 0。

2. Standard Form and Identifying Coefficients | 标准形式与系数的确定

Before solving, write the equation in standard form:

求解之前，应先把方程写成标准形式：

ax² + bx + c = 0

Then identify the coefficients a, b and c. For 5x² − 3x + 2 = 0, we have a = 5, b = −3 and c = 2. If terms are on both sides, rearrange and combine like terms first. For example, 2x + x² = 6 becomes x² + 2x − 6 = 0.

然后确定系数 a、b、c。例如 5x² − 3x + 2 = 0 中 a = 5，b = −3，c = 2。如果项分布在等号两边，需要先移项、合并同类项。例如 2x + x² = 6 可化为 x² + 2x − 6 = 0。

• A term like −x has coefficient −1. / 像 −x 这样的项系数为 −1。
• If there is no x term, then b = 0. / 如果没有 x 项，那么 b = 0。
• If there is no constant term, then c = 0. / 如果没有常数项，那么 c = 0。

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