📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Mastering the different methods of solving them will not only boost your exam score but also build a strong foundation for advanced mathematics.
二次方程是 IGCSE 数学中考查频率最高的考点之一。掌握求解二次方程的不同方法，不仅能直接提升考试成绩，也能为后续高阶数学打下坚实基础。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The highest power of the variable x is 2, which is why it is called "quadratic."
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"方程。
ax² + bx + c = 0 (a ≠ 0)
For example: 2x² - 5x + 3 = 0 is a quadratic equation. Here a = 2, b = -5, and c = 3.
例如：2x² - 5x + 3 = 0 是一个二次方程，其中 a = 2，b = -5，c = 3。
2. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic expression can be factored into two linear factors. The key principle is: if the product of two expressions equals zero, then at least one of them must be zero.
当二次式可以分解为两个一次因式时，因式分解法往往是最快捷的方法。其核心原理是：若两个表达式的乘积为零，则其中至少有一个为零。
If A × B = 0, then A = 0 or B = 0
Steps to solve by factorisation:
因式分解法的求解步骤：
- Write the equation in the form ax² + bx + c = 0 (make sure the right side is 0).
- 将方程写成 ax² + bx + c = 0 的形式（确保右边为 0）。
- Factorise the left-hand side into two brackets.
- 将左边因式分解为两个括号相乘的形式。
- Set each bracket equal to 0 and solve for x.
- 令每个括号等于 0，分别解出 x。
Example: Solve x² - 5x + 6 = 0.
示例：解方程 x² - 5x + 6 = 0。
x² - 5x + 6 = (x - 2)(x - 3) = 0
So x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
因此 x - 2 = 0 或 x - 3 = 0，得到 x = 2 或 x = 3。
3. Perfect Squares and Difference of Two Squares | 完全平方与平方差公式
Two special factorisation patterns are very useful: the perfect square trinomial a² ± 2ab + b² = (a ± b)² and the difference of two squares a² - b² = (a + b)(a - b).
两种特殊的因式分解模式非常实用：完全平方三项式 a² ± 2ab + b² = (a ± b)² 和平方差公式 a² - b² = (a + b)(a - b)。
Example 1: x² + 6x + 9 = 0 → (x + 3)² = 0 → x = -3 (a repeated root).
示例 1：x² + 6x + 9 = 0 → (x + 3)² = 0 → x = -3（重根）。
Example 2: 4x² - 25 = 0 → (2x + 5)(2x - 5) = 0 → x = -5/2 or x = 5/2.
示例 2：4x² - 25 = 0 → (2x + 5)(2x - 5) = 0 → x = -5/2 或 x = 5/2。
Recognising these patterns can save you significant time in the exam.
在考试中识别这些模式可以为你节省大量时间。
4. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic equation in the form (x + p)² = q. This method works for any quadratic equation and also helps in finding turning points of parabolas.
配方法将二次方程改写为 (x + p)² = q 的形式。这一方法适用于所有二次方程，还能帮助求抛物线的顶点坐标。
Steps for solving ax² + bx + c = 0 by completing the square:
用配方法解 ax² + bx + c = 0 的步骤：
- If a ≠ 1, divide the entire equation by a first.
- 若 a ≠ 1，先将方程两边同时除以 a。
- Move the constant term c to the right side.
- 将常数项 c 移到等号右边。
- Add (b/2)² to both sides.
- 两边同时加上 (b/2)²。
- Write the left side as a perfect square and solve.
- 将左边写成完全平方式并求解。
Example: Solve x² + 6x - 7 = 0.
示例：解方程 x² + 6x - 7 = 0。
x² + 6x + 9 = 16 ⇒ (x + 3)² = 16
x + 3 = ±4 ⇒ x = 1 or x = -7
The symbol ± means "plus or minus", giving two possible solutions.
符号 ± 表示"正或负"，由此得到两个可能的解。
5. The Quadratic Formula | 求根公式
The quadratic formula provides a direct way to solve any quadratic equation. It is especially useful when factorisation is difficult or impossible.
求根公式为解任意二次方程提供了直接途径。当因式分解困难或无法分解时，它尤为有用。
x = (-b ± √b² - 4ac) / 2a
To use the formula, substitute the values of a, b, and c from the equation ax² + bx + c = 0 into the formula.
使用时，将方程 ax² + bx + c = 0 中的 a、b、c 值代入公式即可。
Example: Solve 2x² - 3x - 2 = 0 using the formula.
示例：用求根公式解方程 2x² - 3x - 2 = 0。
Here a = 2, b = -3, c = -2. Substituting:
其中 a = 2，b = -3，c = -2。代入得：
x = (3 ± √9 + 16) / 4 = (3 ± 5) / 4
Therefore x = 2 or x = -1/2.
因此 x = 2 或 x = -1/2。
6. The Discriminant | 判别式
The expression b² - 4ac is called the discriminant, often denoted by the Greek letter Δ (delta). It determines the number and type of roots without solving the equation.
表达式 b² - 4ac 称为判别式，常用希腊字母 Δ（德尔塔）表示。它可以在不解方程的情况下判断根的个数和类型。
| Discriminant (Δ) | Number and Type of Roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots (two complex roots) |
| 判别式 (Δ) | 根的个数与类型 |
| Δ > 0 | 两个不相等的实数根 |
| Δ = 0 | 一个重根（两个相等的实数根） |
| Δ < 0 | 无实数根（两个复数根） |
Example: Determine the nature of roots of x² - 4x + 5 = 0.
示例：判断方程 x² - 4x + 5 = 0 的根的性质。
Δ = (-4)² - 4(1)(5) = 16 - 20 = -4 < 0, so the equation has no real roots.
Δ = (-4)² - 4(1)(5) = 16 - 20 = -4 < 0，因此该方程无实数根。
7. Which Method Should You Choose? | 如何选择解法？
A common question in exams is: "Solve the equation 2x² + 5x - 3 = 0." Your choice of method depends on the equation itself.
考试中常见的问题是："解方程 2x² + 5x - 3 = 0。" 方法的选择取决于方程本身。
- Check if the equation can be factorised first — this is usually the quickest.
- 首先检查方程能否因式分解——这通常是最快捷的。
- If the coefficient of x is even, completing the square may be neat.
- 如果 x 的系数为偶数，配方法可能更简洁。
- The quadratic formula always works — use it when other methods are not obvious.
- 求根公式总是可行的——当其他方法不明显时使用它。
Regardless of the method you choose, always check your answers by substituting them back into the original equation.
无论选择哪种方法，都要将答案代回原方程进行检验。
8. Solving by Taking Square Roots | 开平方法
erm (bx = 0), it can be written as ax² + c = 0 and solved directly by taking square roots.
当二次方程没有一次项（bx = 0）时，可写成 ax² + c = 0，通过直接开平方求解。
Example: Solve 3x² - 27 = 0.
示例：解方程 3x² - 27 = 0。
x² = 9 ⇒ x = ±3
Note that x² = 9 has two solutions: x = 3 and x = -3. Never forget the negative root!
注意 x² = 9 有两个解：x = 3 和 x = -3。千万不要漏掉负根！
9. Word Problems Leading to Quadratic Equations | 二次方程应用题
Many exam questions present real-world scenarios that require setting up a quadratic equation. The key steps are:
许多考题会给出实际情境，需要列出一个二次方程。关键步骤如下：
- Read the problem carefully and identify the unknown quantity.
- 仔细阅读题目，确定未知量。
- Form a quadratic equation based on the given conditions.
- 根据已知条件列出二次方程。
- Solve the equation using an appropriate method.
- 用合适的方法解方程。
- Check whether both solutions make sense in the context.
- 检查两个解在题意中是否都合理。
Example: The area of a rectangle is 36 cm². Its length is 5 cm more than its width. Find the width.
示例：一个长方形的面积为 36 cm²，长比宽多 5 cm。求宽。
Let the width be x cm. Then the length is (x + 5) cm. The area gives:
设宽为 x cm，则长为 (x + 5) cm。由面积可得：
x(x + 5) = 36 ⇒ x² + 5x - 36 = 0
(x + 9)(x - 4) = 0 ⇒ x = -9 or x = 4
Since width cannot be negative, x = 4 cm.
由于宽度不能为负，因此 x = 4 cm。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Here are the most common mistakes students make with quadratic equations, along with tips to avoid them.
以下是学生在处理二次方程时最常犯的错误，以及相应的避免技巧。
| Common Mistake | 常见错误 | How to Avoid | 如何避免 |
| Forgetting to set the equation to zero before factorising | Always rearrange to ax² + bx + c = 0 first |
| 因式分解前忘记将方程化为零 | 先整理成 ax² + bx + c = 0 的形式 |
| Losing the negative root when taking square roots | Write ± explicitly before solving |
| 开平方时漏掉负根 | 在求解前明确写出 ± 符号 |
| Sign errors when substituting into the formula | Use brackets when substituting negative values |
| 代入公式时出现符号错误 | 代负数时加上括号 |
Show all your working out. The quadratic formula can earn you full marks even if your final answer is slightly wrong — but only if the substitution is visible.
务必写出完整的解题过程。即使最终答案有误，只要代入过程清晰可见，求根公式法仍可能获得满分。
11. Summary | 总结
Quadratic equations are a vital part of IGCSE Mathematics. Whether you factorise, complete the square, or use the formula, the goal is the same: find all values of x that satisfy the equation.
二次方程是 IGCSE 数学的重要组成部分。无论你使用因式分解、配方还是求根公式，目标都是一样的：找出满足方程的所有 x 值。
Remember the flowchart of strategies:
记住以下策略流程图：
- Check if it is in the form (x + p)² = q → take square roots.
- 检查能否化为 (x + p)² = q 的形式 → 直接开平方。
- Check if it can be factorised with integers → factorise.
- 检查能否用整数进行因式分解 → 分解因式。
- Otherwise → use the quadratic formula or complete the square.
- 否则 → 使用求根公式或配方法。
Practice with a wide variety of equations, and you will build the confidence to tackle any quadratic question that appears in your exam.
多做不同类型的方程练习，你将建立信心，从容应对考试中的任何二次方程题目。
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