Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in almost every exam paper, either as direct questions or as part of problem solving, graphs and coordinate geometry. Mastering the different methods of solving quadratics is essential for achieving a high grade, whether you are following Edexcel, AQA, OCR or Cambridge IGCSE specifications.

一元二次方程是 IGCSE 数学中最重要的考点之一。几乎每次考试都会出现，无论是直接考查，还是作为解题、函数图像和坐标几何的一部分。无论你学习的是 Edexcel、AQA、OCR 还是 Cambridge IGCSE 大纲，掌握解一元二次方程的不同方法都是取得高分的关键。

1. What Is a Quadratic Equation? | 什么是一元二次方程

are constants, and a ≠ 0. The highest power of the variable x is 2, which is why it is called 'quadratic'. The word 'quadratic' comes from the Latin 'quadratus', meaning square, because the variable is squared.

一元二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此称为"二次"。"二次"一词源自拉丁语 "quadratus"，意为"平方"，因为变量被平方了。

For example:

例如：

3x² + 5x - 2 = 0, x² - 7x + 12 = 0, 2x² - 8 = 0

not quadratic. To solve a quadratic equation means to find the values of x that make the equation true. These values are called the roots or solutions of the equation. A quadratic equation can have two distinct roots, two equal roots, or no real roots, depending on the values of a, b and c.

如果 a = 0，方程就变成了一次方程，而不是二次方程。解一元二次方程就是求出使方程成立的 x 的值，这些值称为方程的根或解。根据 a、b、c 的取值不同，一元二次方程可以有两个不相等的根、两个相等的根，或者没有实数根。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic has simple integer roots. The idea is to write the quadratic expression as a product of two linear factors, then set each factor equal to zero using the zero product property.

当二次方程具有简单整数根时，因式分解通常是最快的方法。其思路是把二次表达式写成两个一次因式的乘积，然后利用零乘积性质令每个因式分别等于零。

Step 1: Write the equation in the form ax² + bx + c = 0.

第一步：将方程写成 ax² + bx + c = 0 的形式。

Step 2: Factorise the left-hand side completely.

第二步：对左边进行彻底因式分解。

Step 3: Set each factor to zero and solve for x.

第三步：令每个因式等于零并求解 x。

Example: Solve x² - 7x + 12 = 0.

例题：解方程 x² - 7x + 12 = 0。

add to -7. These are -3 and -4, because (-3) × (-4) = 12 and (-3) + (-4) = -7. Therefore:

我们需要找到两个数，乘积为 12，和为 -7。这两个数是 -3 和 -4，因为 (-3) × (-4) = 12，且 (-3) + (-4) = -7。因此：

(x - 3)(x - 4) = 0

So x - 3 = 0 or x - 4 = 0, which gives x = 3 or x = 4.

所以 x - 3 = 0 或 x - 4 = 0，解得 x = 3 或 x = 4。

-6 and add to b = 5: these are 6 and -1. Split the middle term and factorise by grouping:

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