📚 Solving Quadratic Equations: A Complete IGCSE Guide | 解二次方程：IGCSE 完整指南
Quadratic equations appear frequently in IGCSE Mathematics, from factorisation to the quadratic formula and completing the square. This guide walks you through every method step by step, with exam-style examples and common pitfalls clearly explained.
二次方程在 IGCSE 数学中频繁出现，从因式分解到求根公式再到配方法，都是核心考点。本指南将逐步讲解每一种解法，并结合考试题型和常见易错点进行清晰说明。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable is 2, which gives the graph a distinctive U-shaped curve called a parabola.
二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量的最高次数为 2，这使得其图像呈现独特的 U 形曲线，称为抛物线。
For example, 2x² + 3x − 5 = 0 is quadratic, while x² can never be missing if the equation is truly quadratic. If a = 0, the equation becomes linear, not quadratic.
例如，2x² + 3x − 5 = 0 是二次方程；如果真正的二次方程中 x² 项缺失，那就不再是二次方程。若 a = 0，方程退化为线性方程而非二次方程。
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 rewrite ax² + bx + c as a product of two linear expressions, then set each factor to zero.
当二次方程具有简单的整数根时，因式分解通常是最快的方法。其核心思想是将 ax² + bx + c 重写为两个一次表达式的乘积，然后将每个因式分别设为零。
Example: Solve x² − 5x + 6 = 0.
示例： 解方程 x² − 5x + 6 = 0。
We look for two numbers that multiply to +6 and add to −5. These numbers are −2 and −3, so:
我们需要找到两个数，它们相乘等于 +6，相加等于 −5。这两个数是 −2 和 −3，因此：
(x − 2)(x − 3) = 0
Setting each bracket to zero gives x = 2 or x = 3. Always expand your brackets to check the answer.
令每个括号等于零，得到 x = 2 或 x = 3。务必展开括号验证答案是否正确。
3. The Difference of Two Squares | 平方差公式
A special factorisation case occurs when the quadratic has the form x² − a² = 0. This factorises directly as (x − a)(x + a) = 0.
当二次方程具有 x² − a² = 0 的形式时，会出现一种特殊的因式分解情况，可以直接分解为 (x − a)(x + a) = 0。
Example: Solve x² − 49 = 0.
示例： 解方程 x² − 49 = 0。
(x − 7)(x + 7) = 0
Therefore x = 7 or x = −7. This method also works with other square numbers and even algebraic expressions.
因此 x = 7 或 x = −7。此方法同样适用于其他平方数甚至代数表达式。
4. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful not only for solving equations but also for finding the vertex of a parabola.
配方法将二次方程改写为 a(x + p)² + q 的形式。这不仅有助于解方程，还能用于求抛物线的顶点。
Example: Solve x² + 6x + 1 = 0 by completing the square.
示例： 用配方法解 x² + 6x + 1 = 0。
First, halve the coefficient of x (which is 6) to get 3. Write (x + 3)², which expands to x² + 6x + 9. Since we only have +1, we must subtract 8:
首先，将 x 的系数 6 取半得到 3。写出 (x + 3)²，展开为 x² + 6x + 9。由于原方程只有 +1，我们需要减去 8：
(x + 3)² − 8 = 0
(x + 3)² = 8
x + 3 = ±√8
x = −3 ± 2√2
The ± symbol indicates two solutions: x = −3 + 2√2 and x = −3 − 2√2.
± 符号表示两个解：x = −3 + 2√2 和 x = −3 − 2√2。
5. The Quadratic Formula | 求根公式
The quadratic formula is a universal method that works for any quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0, the solutions are given by:
求根公式是一种通用方法，适用于任何二次方程，即使因式分解困难或无法分解也同样有效。对于 ax² + bx + c = 0，解由以下公式给出：
x = (−b ± √(b² − 4ac)) ÷ (2a)
This formula is derived from completing the square and is provided in the IGCSE formula sheet. You must substitute a, b and c carefully, paying attention to negative signs.
该公式由配方法推导而来，并已在 IGCSE 公式表中给出。代入 a、b、c 时务必小心，特别注意负号。
6. Using the Discriminant | 判别式的应用
The part of the quadratic formula under the square root, b² − 4ac, is called the discriminant. It tells us how many real roots a quadratic equation has.
求根公式中根号下的部分 b² − 4ac 称为判别式，它告诉我们二次方程有多少个实数根。
| Discriminant 判别式 | Nature of Roots 根的性质 |
| b² − 4ac > 0 | Two distinct real roots 两个不同实数根 |
| b² − 4ac = 0 | One repeated real root 一个重根 |
| b² − 4ac < 0 | No real roots 无实数根 |
If the discriminant is negative, the equation has complex or imaginary roots, which are not tested on the IGCSE core syllabus.
若判别式为负，方程具有复数根或虚根，这不在 IGCSE 核心大纲的考查范围内。
7. Solving Word Problems | 应用题解法
Many IGCSE questions present quadratic equations in context. For example, the area of a rectangle is given, and you must find its dimensions.
许多 IGCSE 题目以实际情境呈现二次方程。例如，已知矩形的面积，要求求其边长。
Example: A rectangle has length (x + 4) cm and width x cm. Its area is 45 cm². Find x.
示例： 一个矩形的长为 (x + 4) 厘米，宽为 x 厘米。它的面积为 45 平方厘米。求 x。
x(x + 4) = 45
x² + 4x − 45 = 0
(x + 9)(x − 5) = 0
This gives x = −9 or x = 5. Since a length cannot be negative, we take x = 5 cm. The dimensions are 5 cm and 9 cm.
这给出 x = −9 或 x = 5。由于长度不能为负，我们取 x = 5 厘米。矩形的尺寸为 5 厘米和 9 厘米。
8. Solving by Using the Graph | 图像解法
A quadratic equation can also be solved graphically. The roots of ax² + bx + c = 0 are the x-coordinates where the parabola crosses the x-axis.
二次方程也可以通过图像来解。ax² + bx + c = 0 的根是抛物线与 x 轴交点的 x 坐标。
If the graph does not cross the x-axis, the equation has no real solutions. If it touches the axis at one point, there is exactly one repeated root.
如果图像不与 x 轴相交，则方程无实数解。如果图像与 x 轴相切于一点，则恰好有一个重根。
You may also need to draw a suitable straight line on a known curve to solve a more complex equation, such as x² − 2x − 3 = 0 from the curve y = x² − 2x − 3 by reading off the x-intercepts.
有时需要先在已知曲线上画一条合适的直线，从而求解更复杂的方程。例如从曲线 y = x² − 2x − 3 中读取与 x 轴的交点来解 x² − 2x − 3 = 0。
9. Common Mistakes to Avoid | 常见易错点
One of the most frequent errors is forgetting to set the equation to zero before factorising. For example, solving x² + 3x = 4 directly by factorising x(x + 3) = 4 will lead to an incorrect answer.
最常见的错误之一是在因式分解前忘记将方程归零。例如，直接将 x² + 3x = 4 分解为 x(x + 3) = 4 会导致错误答案。
- Always rearrange to ax² + bx + c = 0 first | 务必先将方程整理为 ax² + bx + c = 0
- Check your factorisation by expanding | 通过展开来验算因式分解是否正确
- When using the formula, write down a, b and c before substituting | 使用求根公式前，先写出 a、b、c 再代入
- Do not forget the ± sign in the quadratic formula | 不要忘记求根公式中的 ± 符号
- Read word problems carefully to decide whether to reject a negative root | 仔细阅读应用题，判断是否舍弃负根
10. Practice Questions | 练习题
Try these questions on your own before checking the answers below. They cover all the methods discussed in this guide.
请先独立尝试以下题目，再核对下方答案。这些题目涵盖本指南讨论的所有方法。
- Solve x² − 7x + 10 = 0 | 解 x² − 7x + 10 = 0
- Solve x² + 8x − 20 = 0 by completing the square | 用配方法解 x² + 8x − 20 = 0
- Use the quadratic formula to solve 2x² + 5x − 3 = 0 | 用求根公式解 2x² + 5x − 3 = 0
- Determine the nature of roots for x² + 4x + 5 = 0 | 判断 x² + 4x + 5 = 0 根的性质
Answers: x = 2 or 5; x = −10 or 2; x = 0.5 or −3; discriminant = −4, no real roots.
答案：x = 2 或 5；x = −10 或 2；x = 0.5 或 −3；判别式 = −4，无实数根。
11. Summary of Methods | 方法总结
Choosing the right method can save time in the exam. Factorisation is quickest for simple equations, completing the square is best for finding the vertex, and the quadratic formula always works.
在考试中选择合适的方法可以节省时间。对于简单方程，因式分解最快；对于求顶点，配方法最佳；而求根公式则始终有效。
| Method 方法 | Best Used When 适用情况 |
| Factorisation 因式分解 | Simple integer roots 简单整数根 |
| Completing the square 配方法 | Finding vertex or when a = 1 求顶点或 a = 1 时 |
| Quadratic formula 求根公式 | Any quadratic, especially non-factorisable 适用于任何方程，特别是无法因式分解的 |
| Graphical method 图像法 | When a graph is given 当已给出图像时 |
Make sure you practise all four methods until you can apply them confidently under time pressure.
务必练习全部四种方法，直到你能够在时间压力下自信地运用为止。
12. Final Exam Tips | 考前提分要点
In the IGCSE exam, always show your working clearly. Even if your final answer is wrong, you can still gain method marks for correct steps.
在 IGCSE 考试中，务必清晰展示你的解题过程。即使最终答案错误，正确的步骤仍然可以获得步骤分。
Check whether the question asks for exact values or decimal answers. If it says “correct to 2 decimal places”, use that precision at the very end of your working.
注意题目要求精确值还是小数答案。如果题目要求”保留到两位小数”，请在计算的最后一步再取近似值。
Remember that a quadratic equation can have two, one or zero real solutions. Look at the discriminant to verify your answer makes sense.
记住二次方程可能有两个、一个或零个实数解。通过判别式可以验证你的答案是否合理。
With consistent practice and a clear understanding of each method, solving quadratic equations will become one of the most reliable skills in your IGCSE Mathematics toolkit.
通过持续练习和深入理解每种方法，解二次方程将成为你 IGCSE 数学考试中最可靠的技能之一。
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