📚 Solving Quadratic Equations: Factorisation and the Formula | 解二次方程：因式分解与求根公式
Quadratics are among the most important topics in IGCSE Mathematics. In this revision guide, you will learn how to solve quadratic equations using factorisation, completing the square, and the quadratic formula, with worked examples, common mistakes, and practice questions.
二次方程是 IGCSE 数学中最重要的内容之一。本复习指南将讲解如何用因式分解、配方法与求根公式解二次方程，并配有典型例题、常见错误和练习题。
1. What Is a Quadratic Equation? | 什么是二次方程
A quadratic equation is an equation in which the highest power of the unknown is 2. In its standard form, it is written as
二次方程是未知数最高次数为 2 的方程。其标准形式为
ax² + bx + c = 0, a ≠ 0
Here, a, b and c are constants. If a = 0, the x² term disappears and the equation becomes linear, so it is no longer quadratic. The values of b and c may be zero, but a cannot be zero.
其中 a、b、c 是常数。如果 a = 0，x² 项就会消失，方程变成一次方程，因此不再是二次方程。b 和 c 可以为 0，但 a 不能为 0。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving a quadratic equation, you should rearrange it into the standard form ax² + bx + c = 0. This makes it easier to identify a, b and c, whether you are factorising or using the formula.
在解二次方程之前，应先将其整理成标准形式 ax² + bx + c = 0。这样无论使用因式分解还是求根公式，都能更容易地确定 a、b、c。
| Symbol | Meaning | 中文 |
| a | coefficient of x² | 二次项系数 |
| b | coefficient of x | 一次项系数 |
| c | constant term | 常数项 |
| root / solution | a value of x that makes the equation equal 0 | 根 / 解 |
3. Solving by Factorisation | 用因式分解解题
If a quadratic expression can be written as a product of two brackets, you can solve it using the zero product rule. This rule says: if A and B are real numbers and AB = 0, then A = 0 or B = 0.
如果一个二次表达式可以写成两个括号相乘的形式，就可以利用零乘积性质来求解。该性质说明：如果 A、B 是实数且 AB = 0，那么 A = 0 或 B = 0。
If AB = 0, then A = 0 or B = 0
To factorise x² + bx + c, look for two numbers whose sum is b and whose product is c. For example, to solve x² + 5x + 6 = 0, find two numbers with sum 5 and product 6. The numbers are 2 and 3.
要分解 x² + bx + c，需要找到两个数，使它们的和为 b，积为 c。例如，解 x² + 5x + 6 = 0 时，找两个和为 5、积为 6 的数，这两个数是 2 和 3。
4. Worked Example: Factorisation | 因式分解例题
Solve x² − 5x + 6 = 0.
解方程 x² − 5x + 6 = 0。
We need two numbers whose sum is −5 and whose product is 6. The numbers are −2 and −3. Therefore the equation factorises as:
我们需要找到两个数，使它们的和为 −5，积为 6。这两个数是 −2 和 −3。因此方程可以分解为：
x² − 5x + 6 = (x − 2)(x − 3) = 0
Using the zero product rule, either x − 2 = 0 or x − 3 = 0. So the solutions are x = 2 and x = 3.
根据零乘积性质，x − 2 = 0 或 x − 3 = 0。所以方程的解是 x = 2 和 x = 3。
Check: when x = 2, 2² − 5(2) + 6 = 4 − 10 + 6 = 0. When x = 3, 3² − 5(3) + 6 = 9 − 15 + 6 = 0.
检验：当 x = 2 时，2² − 5×2 + 6 = 4 − 10 + 6 = 0。当 x = 3 时，3² − 5×3 + 6 = 9 − 15 + 6 = 0。
5. Solving by Completing the Square | 配方法
Completing the square is a method that rewrites part of the quadratic expression as a perfect square plus a constant. For x² + bx, the rule is:
配方法将二次表达式的一部分改写为完全平方加上一个常数。对于 x² + bx，公式为：
x² + bx = (x + b/2)² − (b/2)²
For example, solve x² + 6x + 2 = 0. Since b = 6, half of b is 3. So x² + 6x = (x + 3)² − 9. The equation becomes:
例如，解 x² + 6x + 2 = 0。因为 b = 6，b 的一半是 3，所以 x² + 6x = (x + 3)² − 9。原方程变为：
(x + 3)² − 9 + 2 = 0
(x + 3)² = 7
Taking the square root of both sides gives x + 3 = ±√7, so the exact solutions are x = −3 + √7 and x = −3 − √7.
两边开平方得 x + 3 = ±√7，因此精确解为 x = −3 + √7 和 x = −3 − √7。
6. The Quadratic Formula | 求根公式
The quadratic formula is a universal method for solving any quadratic equation written in the form ax² + bx + c = 0. It is especially useful when the quadratic cannot be factorised easily.
求根公式是解任意标准形式 ax² + bx + c = 0 二次方程的通用方法。当二次式不容易因式分解时，它尤其有用。
x = (−b ± √(b² − 4ac)) / (2a)
To use the formula, first identify a, b and c. Then substitute them into the formula and simplify. The ± symbol means that one root uses addition and the other uses subtraction.
使用公式时，先确定 a、b、c，然后代入公式并化简。符号 ± 表示一个根用加法，另一个根用减法。
7. The Discriminant and Roots | 判别式与根
The expression inside the square root, b² − 4ac, is called the discriminant. It tells you how many real roots the quadratic equation has without solving it completely.
根号内的表达式 b² − 4ac 称为判别式。它可以在不完全解方程的情况下，判断二次方程有多少个实数根。
| Discriminant | Number of Real Roots | 中文 |
| b² − 4ac > 0 | two distinct real roots | 两个不同的实数根 |
| b² − 4ac = 0 | one repeated real root | 一个重根 |
| b² − 4ac < 0 | no real roots | 没有实数根 |
8. Worked Example: The Formula | 求根公式例题
Solve 2x² + 3x − 2 = 0 using the quadratic formula.
用求根公式解方程 2x² + 3x − 2 = 0。
Here a = 2, b = 3 and c = −2. Substituting into the formula:
这里 a = 2，b = 3，c = −2。代入公式：
Δ = 3² − 4 × 2 × (−2) = 9 + 16 = 25
x = (−3 ± √25) / (2 × 2) = (−3 ± 5) / 4
Using the plus sign: x = (−3 + 5)/4 = 2/4 = 1/2. Using the minus sign: x = (−3 − 5)/4 = −8/4 = −2. Therefore the solutions are x = 1/2 and x = −2.
取加号时：x = (−3 + 5)/4 = 2/4 = 1/2。取减号时：x = (−3 − 5)/4 = −8/4 = −2。因此方程的解是 x = 1/2 和 x = −2。
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