📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are among the most frequently tested topics in IGCSE Mathematics. They appear in both Paper 2 and Paper 4 and may be examined as pure algebra, graph sketching, or word problems. This revision guide walks you through every essential method step by step, with worked examples in both English and Chinese.
二次方程是 IGCSE 数学中考查频率最高的知识点之一，在 Paper 2 和 Paper 4 中都会出现，并可能以纯代数、图像或应用题的形式考查。本复习指南将逐步讲解所有关键方法，并配中英双语例题。
1. What Is a Quadratic Equation? | 什么是二次方程
A quadratic equation is an equation of degree 2, meaning the highest power of the unknown variable is exactly 2. Every quadratic equation can be written in the general form:
二次方程是次数为 2 的方程，意思是未知数的最高次数恰好为 2。每个二次方程都可以写成一般形式：
ax² + bx + c = 0, where a ≠ 0
Here a, b and c are constants, and x is the unknown. The condition a ≠ 0 is essential: if a = 0, the equation becomes linear, not quadratic.
这里 a、b、c 是常数，x 是未知数。条件 a ≠ 0 是必要的：如果 a = 0，方程就变成了一次方程，而不是二次方程。
For example, x² + 5x + 6 = 0 is quadratic, while 2x + 1 = 0 is linear and x³ − 2x = 0 is cubic. Only the highest power 2 gives the equation its name.
例如，x² + 5x + 6 = 0 是二次方程，而 2x + 1 = 0 是一次方程，x³ − 2x = 0 是三次方程。只有最高次数 2 决定了方程的名称。
2. The Standard Form | 标准形式
Before solving, a quadratic equation must be rearranged so that all terms are on one side of the equals sign and the right-hand side is zero. For example:
解题之前，必须先将二次方程整理为所有项都在等号一侧、右侧为 0 的形式。例如：
2x² + 3 = 5x → 2x² − 5x + 3 = 0
In this example, a = 2, b = −5 and c = 3. Note that b or c may be zero; for instance, x² − 9 = 0 has b = 0, while 3x² − 4x = 0 has c = 0.
在这个例子中，a = 2，b = −5，c = 3。注意 b 或 c 可以为零；例如 x² − 9 = 0 中 b = 0，而 3x² − 4x = 0 中 c = 0。
Always check that the coefficient of x² is not zero, and if it is negative, it is often helpful to multiply the whole equation by −1 to make a positive.
务必检查 x² 的系数不为零；如果系数为负，通常可以将整个方程乘以 −1，使 a 变成正数，便于计算。
3. Solving by Factorisation | 因式分解法
Factorisation is the fastest method when the quadratic expression factorises neatly. The key idea is the zero product property: if the product of two expressions is zero, then at least one of them must be zero.
当二次式可以顺利分解时，因式分解法是最快捷的方法。核心依据是零积性质：若两个因式的乘积为零，则至少有一个因式为零。
Follow these steps:
解题步骤如下：
- Rearrange into standard form ax² + bx + c = 0.
- Factorise the left-hand side into two brackets.
- Set each bracket equal to zero and solve.
- 先将方程整理成标准形式 ax² + bx + c = 0。
- 将左侧分解为两个括号的乘积。
- 令每个括号等于零，分别解出答案。
Worked example: Solve x² + 7x + 10 = 0.
例题：解方程 x² + 7x + 10 = 0。
x² + 7x + 10 = (x + 2)(x + 5) = 0
Hence x + 2 = 0 or x + 5 = 0, so x = −2 or x = −5.
因此 x + 2 = 0 或 x + 5 = 0，所以 x = −2 或 x = −5。
Another common type is the difference of two squares. For example, x² − 9 = 0 becomes (x − 3)(x + 3) = 0, giving x = 3 or x = −3. Always look for this pattern first.
另一种常见题型是平方差公式。例如，x² − 9 = 0 可化为 (x − 3)(x + 3) = 0，得到 x = 3 或 x = −3。做题时应优先寻找这种模式。
4. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This method is especially useful when the equation cannot be factorised, and it also helps in sketching graphs and finding turning points.
配方法将二次式改写为 a(x + p)² + q 的形式。当方程无法因式分解时，这个方法尤其有用，同时也有助于绘制图像和求顶点坐标。
For a quadratic x² + bx, we use the identity:
对于二次式 x² + bx，我们使用恒等式：
x² + bx = (x + b/2)² − (b/2)²
Worked example: Solve x² + 6x + 2 = 0.
例题：解方程 x² + 6x + 2 = 0。
(x + 3)² − 9 + 2 = 0 → (x + 3)² = 7
x + 3 = ±√7 → x = −3 ± √7
Thus the two solutions are x = −3 + √7 and x = −3 − √7. In IGCSE, you may be asked to leave the answer in surd form exactly like this.
因此两个解为 x = −3 + √7 和 x = −3 − √7。在 IGCSE 考试中，可能需要像这样以根式形式保留精确答案。
5. The Quadratic Formula | 二次公式
The quadratic formula solves any quadratic equation, whether or not it factorises. It is derived from completing the square and is included in the IGCSE formula sheet, but you must know how to substitute correctly.
二次公式可以解任何二次方程，无论能否因式分解。它由配方法推导而来，IGCSE 公式表中会给出，但你必须掌握如何正确代入。
x = (−b ± √(b² − 4ac)) / 2a
Worked example: Solve 2x² + 3x − 2 = 0 using the formula.
例题：用二次公式解 2x² + 3x − 2 = 0。
Here a = 2, b = 3 and c = −2. Substitute:
这里 a = 2，b = 3，c = −2。代入得：
x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4 = (−3 ± 5) / 4
So x = (−3 + 5) / 4 = 0.5 or x = (−3 − 5) / 4 = −2. The solutions are x = 0.5 and x = −2.
因此 x = (−3 + 5) / 4 = 0.5，或 x = (−3 − 5) / 4 = −2。解为 x = 0.5 和 x = −2。
When substituting, be very careful with negative signs, especially when c is negative. Write out every step to avoid arithmetic errors.
代入时，要特别注意负号，尤其是当 c 为负数时。写出每一步可以避免计算错误。
6. The Discriminant | 判别式
The expression inside the square root, b² − 4ac, is called the discriminant and is denoted by Δ. It tells us how many real roots a quadratic equation has without solving it fully.
根号内的表达式 b² − 4ac 称为判别式，记作 Δ。它可以在不完全求解的情况下告诉我们二次方程有多少个实数根。
Δ = b² − 4ac
There are three cases:
共有三种情况：
- If Δ > 0: the equation has two distinct real roots.
- If Δ = 0: the equation has one repeated real root.
- If Δ < 0: the equation has no real roots.
- 若 Δ > 0：方程有两个不同的实数根。
- 若 Δ = 0：方程有一个重根（两个相等的实数根）。
- 若 Δ < 0：方程没有实数根。
Example: For x² + 4x + 5 = 0, we have Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4, which is negative, so there are no real roots. The graph of y = x² + 4x + 5 never crosses the x-axis.
示例：对于 x² + 4x + 5 = 0，Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4，为负数，因此没有实数根。y = x² + 4x + 5 的图像与 x 轴无交点。
7. The Graph of a Quadratic Function | 二次函数图像
The graph of y = ax² + bx + c is always a parabola. If a > 0, the parabola opens upwards forming a U shape; if a < 0, it opens downwards forming an upside-down U shape.
函数 y = ax² + bx + c 的图像始终是抛物线。当 a > 0 时，抛物线开口向上，呈 U 形；当 a < 0 时，开口向下，呈倒 U 形。
The solutions of the equation ax² + bx + c = 0 are exactly the x-coordinates where the graph crosses the x-axis. These points are called the roots or x-intercepts.
方程 ax² + bx + c = 0 的解正是图像与 x 轴交点的横坐标。这些点称为根或 x 轴截距。
The axis of symmetry of the parabola is the vertical line x = −b/(2a), and the turning point (vertex) lies on this line. Its y-coordinate can be found by substituting x = −b/(2a) into the function.
抛物线的对称轴是竖直线 x = −b/(2a)，顶点就在这条对称轴上。将 x = −b/(2a) 代入函数式即可求得顶点的 y 坐标。
In IGCSE graph questions, you may be asked to sketch the curve, label the roots, and mark the y-intercept at (0, c). Practise drawing parabolas for both positive and negative a.
在 IGCSE 的图像题中，你可能需要画出曲线草图、标出根的位置，并标出 y 轴截距 (0, c)。要多练习绘制 a 为正和为负的抛物线。
8. Word Problems | 应用题
Quadratic equations often arise from geometric or number problems. Setting up the equation correctly is the most important step, and you should always check whether your answers make sense in the context.
二次方程经常来源于几何或数字类应用题。正确列方程是最关键的一步，同时要检查答案是否符合题意。
Worked example: A rectangle has length 5 cm longer than its width. Its area is 84 cm². Find the dimensions.
例题：一个长方形的长比宽长 5 cm，面积为 84 cm²。求长和宽。
Let the width be x cm. Then the length is x + 5 cm. Since area = length × width:
设宽为 x cm，则长为 x + 5 cm。因为面积 = 长 × 宽：
x(x + 5) = 84 → x² + 5x − 84 = 0
(x + 12)(x − 7) = 0
So x = 7 or x = −12. Since a length cannot be negative, x = 7. Therefore the width is 7 cm and the length is 12 cm.
所以 x = 7 或 x = −12。因为长度不能为负，所以 x = 7。因此宽为 7 cm，长为 12 cm。
Always reject negative or unrealistic solutions when the problem describes physical quantities such as lengths, areas, or times.
当问题描述的是长度、面积、时间等物理量时，务必舍去负数或不合理的解。
9. Common Mistakes | 常见错误
Below are the most frequent errors students make in quadratic equation questions. Learn to avoid them and you will gain valuable marks.
以下是最常见的失分陷阱。学会避免这些错误，你就能在考试中拿回不少分数。
- Not writing in standard form: Always rearrange to ax² + bx + c = 0 before factorising or using the formula.
- Losing solutions by dividing by x: If x is a factor, never divide both sides by x; factorise instead.
- Sign errors in substitution: When b or c is negative, substitute carefully with brackets or write the signs explicitly.
- Ignoring the condition a ≠ 0: If a = 0 after rearrangement, the equation is linear, not quadratic.
- 未整理成标准形式：在因式分解或套用公式前，务必先将方程整理成 ax² + bx + c = 0。
- 除以 x 导致丢根：如果 x 是公因子，绝不能两边同除以 x，而应使用因式分解。
- 代入公式时符号错误：当 b 或 c 为负数时，可以加上括号或明确写出符号，小心代入。
- 忽略 a ≠ 0 的条件：若整理后发现 a = 0，则该方程是一次方程，不是二次方程。
10. Summary and Exam Tips | 总结与考试技巧
The table below summarises when to use each solving method. In the exam, choose the most efficient method rather than always using the formula.
下表总结了每种解法的最佳适用场景。在考试中，应选择最高效的方法，而不是每次都套用公式。
| Method | 方法 | When to use | 适用情况 |
| Factorisation | 因式分解法 | When the expression factorises easily or it is a difference of two squares. |
| Completing the square | 配方法 | When you need exact surd answers or need to find the turning point. |
| Quadratic formula | 二次公式 | When factorisation is difficult or not obvious.Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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