📚 Solving Quadratic Equations | 解二次方程
Quadratic equations appear in almost every IGCSE Mathematics exam. They test your algebraic manipulation, logical thinking, and ability to apply abstract methods to real-world problems. In this article, you will learn the essential techniques for solving quadratic equations, with worked examples and common pitfalls to avoid.
二次方程几乎出现在每一场 IGCSE 数学考试中。它考查你的代数运算能力、逻辑思维，以及将抽象方法应用于实际问题的能力。本文将帮助你掌握解二次方程的核心技巧，并提供经典例题与常见易错点分析。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0, and a, b, c are real numbers. The highest power of x is 2, which is why it is called "quadratic".
二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a ≠ 0，a、b、c 为实数。未知数 x 的最高次数为 2，因此称为“二次”方程。
- If a = 0, the equation becomes linear. | 如果 a = 0，方程就变成了一次方程。
- Every quadratic equation has at most two real solutions. | 每个二次方程至多有两个实数解。
ax² + bx + c = 0
2. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic has simple integer factors. You rewrite the equation as a product of two linear expressions, then set each factor equal to zero.
当二次方程有简单的整数因式时，因式分解法通常是最快捷的方法。你将方程改写为两个一次表达式的乘积，然后令每个因式等于零。
Example: Solve x² - 5x + 6 = 0.
示例：解 x² - 5x + 6 = 0。
tiply to 6 and add to -5. These are -2 and -3. Thus (x - 2)(x - 3) = 0, so x = 2 or x = 3.
我们需要找到两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3。因此 (x - 2)(x - 3) = 0，所以 x = 2 或 x = 3。
(x - 2)(x - 3) = 0 ⇒ x = 2 or x = 3
3. Completing the Square | 配方法
thod is particularly useful when the equation cannot be factorised easily, and it also helps in finding the vertex of a parabola.
配方法将方程改写为 (x + p)² = q 的形式。当方程不易因式分解时，此方法非常有用，同时也有助于求抛物线的顶点。
Example: Solve x² + 6x - 7 = 0 by completing the square.
示例：用配方法解 x² + 6x - 7 = 0。
- Move constant: x² + 6x = 7. | 移常数项：x² + 6x = 7。
- Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16. | 两边加上 (6/2)² = 9：x² + 6x + 9 = 16。
- Write as square: (x + 3)² = 16. | 写成完全平方：(x + 3)² = 16。
- Solve: x + 3 = ±4 ⇒ x = 1 or x = -7. | 解得：x + 3 = ±4 ⇒ x = 1 或 x = -7。
(x + 3)² = 16 ⇒ x = 1 or x = -7
4. The Quadratic Formula | 求根公式
The quadratic formula works for every quadratic equation. It is derived from completing the square and is expressed as:
求根公式适用于所有二次方程。它由配方法推导而来，表达为：
x = (-b ± √b² - 4ac) / 2a
Given ax² + bx + c = 0, you substitute the values of a, b and c directly into the formula.
对于 ax² + bx + c = 0，你只需将 a、b、c 的值直接代入公式。
Example: Solve 2x² + 3x - 5 = 0 using the formula.
示例：用公式解 2x² + 3x - 5 = 0。
Here a = 2, b = 3, c = -5. Then b² - 4ac = 9 + 40 = 49, √49 = 7. So x = (-3 ± 7) / 4, giving x = 1 or x = -2.5.
此时 a = 2，b = 3，c = -5。则 b² - 4ac = 9 + 40 = 49，√49 = 7。所以 x = (-3 ± 7) / 4，得到 x = 1 或 x = -2.5。
5. The Discriminant | 判别式
The expression b² - 4ac is called the discriminant. It tells you how many real solutions a quadratic equation has without solving it.
表达式 b² - 4ac 被称为判别式。它无需解方程即可判断二次方程有多少个实数解。
| Discriminant Δ = b² - 4ac | Number of real solutions |
| Δ > 0 | Two distinct real solutions |
| Δ = 0 | One repeated real solution |
| Δ < 0 | No real solutions |
中文对照：Δ > 0 有两个不同实数解；Δ = 0 有一个重根；Δ < 0 没有实数解。
6. Solving by Graphing | 图像法
bx + c and find the x-coordinates where the curve crosses the x-axis. These are the roots.
用图像法解二次方程时，先画出曲线 y = ax² + bx + c，然后找出曲线与 x 轴交点的横坐标。这些横坐标就是方程的根。
If the curve touches the x-axis at one point, there is one repeated root. If it does not touch the x-axis, there are no real roots.
如果曲线与 x 轴相切于一点，则有一个重根；如果曲线不与 x 轴相交，则没有实数根。
This method is useful for estimating solutions, but in exams you usually need exact answers obtained by algebraic methods.
这种方法适用于估算解，但在考试中通常需要用代数方法求出精确答案。
7. Word Problems Involving Quadratics | 二次方程应用题
Many exam problems describe a real-life situation that leads to a quadratic equation. You must define the unknown, set up the equation, solve it, and then interpret both solutions in the context of the problem.
许多考试题目描述一个实际情境，最终会得到一个二次方程。你需要设未知数、建立方程、求解，并在问题情境中解释两个解的含义。
Example: A rectangle has length (x + 3) cm and width x cm. Its area is 70 cm². Find x.
示例：一个长方形的长为 (x + 3) 厘米，宽为 x 厘米。面积为 70 平方厘米。求 x。
Setup: x(x + 3) = 70 ⇒ se: (x + 10)(x - 7) = 0, so x = -10 or x = 7. Since a length cannot be negative, x = 7 cm.
建立方程：x(x + 3) = 70 ⇒ x² + 3x - 70 = 0。因式分解：(x + 10)(x - 7) = 0，所以 x = -10 或 x = 7。由于长度不能为负，因此 x = 7 厘米。
8. Quadratic Functions and Their Graphs | 二次函数及其图像
ays produces a parabola. The sign of a determines whether it opens upward (a > 0) or downward (a < 0).
二次函数 y = ax² + bx + c 的图像总是抛物线。a 的正负决定开口方向：a > 0 时开口向上，a < 0 时开口向下。
- The vertex lies on the axis of symmetry x = -b/(2a). | 顶点位于对称轴 x = -b/(2a) 上。
- The y-intercept is c. | y 轴截距为 c。
- The x-intercepts are the real roots of the equation. | x 轴截距是方程的实数根。
Axis of symmetry: x = -b / (2a)
9. Finding the Vertex by Completing the Square | 用配方法求顶点
en as y = a(x - h)² + k, the vertex is simply (h, k). This form is called the vertex form.
当二次函数写成 y = a(x - h)² + k 时，顶点就是 (h, k)。这种形式称为顶点式。
Example: Express y = x² - 4x + 1 in vertex form.
示例：将 y = x² - 4x + 1 化为顶点式。
Complete the square: y = (x² - 4x + 4) - 4 + 1 = (x - 2)² - 3. So the vertex is (2, -3).
配方：y = (x² - 4x + 4) - 4 + 1 = (x - 2)² - 3。因此顶点为 (2, -3)。
This method is especially important for questions about maximum or minimum values of a quadratic function.
这种方法在求二次函数最大值或最小值的问题中尤其重要。
10. Solving Quadratic Inequalities | 二次不等式
Quadratic inequalities may also appear, such as x² - 5x + 6 > 0. The approach is to first solve the corresponding equation, then test intervals on a number line.
考试中也可能出现二次不等式，例如 x² - 5x + 6 > 0。解法是先求对应方程的根，然后在数轴上检验区间。
2 and x = 3. The parabola opens upward, so the inequality is positive when x < 2 or x > 3.
对于 x² - 5x + 6 = 0，根为 x = 2 和 x = 3。抛物线开口向上，因此不等式大于零在 x < 2 或 x > 3 时成立。
x² - 5x + 6 > 0 ⇒ x < 2 or x > 3
11. Common Mistakes | 常见易错点
Students often lose marks on quadratic equation questions because of small but critical errors. Here are the most frequent ones you should avoid.
学生在解二次方程时常常因为一些微小但关键的失误而丢分。以下是最常见的错误，请务必避免。
- Forgetting to rearrange into standard form before factorising or using the formula. | 在因式分解或使用公式前忘记将方程化为标准形式。
- Miscopying signs: always check whether b or c is negative. | 抄错符号：务必确认 b 或 c 是否为负。
- Dividing both sides by x, which loses the solution x = 0. | 两边同时除以 x，导致丢失解 x = 0。
- Using the discriminant incorrectly: remember Δ = b² - 4ac, not b² - 4a. | 错误使用判别式：请记住 Δ = b² - 4ac，而不是 b² - 4a。
- In word problems, accepting a negative solution without checking whether it makes sense. | 在应用题中，不检查负解是否合理就直接接受。
12. Exam Strategies and Practice | 考试策略与练习
In the Edexcel IGCSE exam, quadratic questions appear in both Paper 1 and Paper 2. Always show your working clearly, because method marks are awarded even if your final answer is wrong.
在 Edexcel IGCSE 考试中，二次方程题目在 Paper 1 和 Paper 2 中都会出现。请务必清晰写出解题过程，因为即使最终答案错误，步骤分仍然可得。
Recommended approach:
推荐方法：
- Write the equation in standard form. | 将方程写成标准形式。
- Try factorisation first. If it is easy, solve immediately. | 先尝试因式分解；如果容易，直接求解。
- If factorisation is not obvious, use completing the square or the quadratic formula. | 如果因式分解不明显，使用配方法或求根公式。
- Use the discriminant to check the number of solutions before solving. | 求解前用判别式检查解的个数。
- For word problems, interpret both algebraic answers and reject any impossible values. | 对于应用题，解释两个代数解并排除不可能的值。
lowing practice question: Solve 3x² - 2x - 1 = 0 by any method. Answer: x = 1 or x = -1/3.
试试下面的练习题：用任何方法解 3x² - 2x - 1 = 0。答案：x = 1 或 x = -1/3。
Mastering quadratics takes regular practice. Use past papers and check every step carefully. You will soon find these questions both predictable and profitable.
掌握二次方程需要定期练习。使用历年真题并仔细检查每一步。你很快会发现这类题目既有规律又容易得分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图