📚 066 Quadratic Equations | 066 二次方程详解
Quadratic equations are a cornerstone of the Edexcel IGCSE Mathematics syllabus. From factorising simple trinomials to applying the quadratic formula and interpreting graphs, mastering this topic is essential for success in both the Foundation and Higher tiers. This article covers all the key methods, common pitfalls, and exam-style applications you need to feel confident when solving any quadratic problem.
二次方程是 Edexcel IGCSE 数学考纲中的基石。从简单的三项式因式分解到使用二次公式和函数图像解读，掌握这个主题对于基础层和进阶层考生都至关重要。本文涵盖所有关键方法、常见易错点和考试题型应用，让你面对任何二次方程问题时都能充满信心。
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 x is 2, which gives the equation its characteristic U-shaped graph called a parabola.
二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数是 2，这使得它的图像呈现出特有的 U 形曲线，称为抛物线。
rearrange terms so that one side equals zero before attempting to solve. Recognising the standard form helps you choose the most efficient solution method.
在 IGCSE 考试中，你可能会遇到不同排列的二次方程，如 x² + 4x - 5 = 0 或 2x² - 8 = 0。在尝试求解前，务必把项移到一边使等式等于零。识别标准形式有助于你选择最有效的解法。
2. Solving by Factorising | 因式分解法求解
ers that multiply to 10 and add to 7: 2 and 5. So the equation becomes (x + 2)(x + 5) = 0.
当二次式可写成两个线性因式的乘积时，因式分解通常是最快的解法。对于如 x² + 7x + 10 = 0 的首一二次式，找出相乘得 10 且相加得 7 的两个数：2 和 5。因此方程变为 (x + 2)(x + 5) = 0。
k your factors by expanding them back to the original expression; a common mistake is to misplace signs.
因式分解后，应用零乘积性质：如果 (x + 2)(x + 5) = 0，则 x + 2 = 0 或 x + 5 = 0。解得 x = -2 和 x = -5。始终通过展开因式检验是否还原为原式；符号放错是常见错误。
atics with a leading coefficient not equal to 1, such as 2x² + 5x - 3 = 0, you can use the 'ac method' middle term, factor by grouping, and obtain (2x - 1)(x + 3) = 0, giving x = 1/2 or x = -3.
对于首项系数不为 1 的二次方程，如 2x² + 5x - 3 = 0，可用“ac 法”或系统试凑。a 与 c 相乘：2 × (-3) = -6。找出乘积为 -6 且和为 5 的两个数：6 和 -1。将中间项拆分，分组分解后得 (2x - 1)(x + 3) = 0，解得 x = 1/2 或 x = -3。
3. The Quadratic Formula | 二次公式
When factorising is difficult or the roots are not integers, the quadratic formula is a reliable tool. The formula is:
x = [-b ± √b² - 4ac] / (2a)
当因式分解困难或根不是整数时，二次公式是一个可靠的工具。公式为：
x = [-b ± √b² - 4ac] / (2a)
3x² - 5x - 2 = 0, a = 3, b = -5, c = -2. Then x = [5 ± √((-5)² - 4×3×(-2))] / (2×3) = [5 ± √25 + 24] / 6 = [5 ± √49] / 6, which simplifies to x = (5 + 7)/6 = 2 or x = (5 - 7)/6 = -1/3.
使用时，先将方程化为 ax² + bx + c = 0 的形式，确定 a、b、c，再代入公式。例如 3x² - 5x - 2 = 0，a = 3, b = -5, c = -2。则 x = [5 ± √((-5)² - 4×3×(-2))] / (2×3) = [5 ± √25 + 24] / 6 = [5 ± √49] / 6，化简得 x = (5+7)/6 = 2 或 x = (5-7)/6 = -1/3。
Remember that the formula works for all quadratics, even those with irrational or complex roots, but at IGCSE level you will mainly see real roots. Always simplify surds where possible, and leave answers in exact form unless asked otherwise.
记住二次公式适用于所有二次方程，即使是有无理数根或复数根的情况，但在 IGCSE 阶段你主要会遇到实数根。尽量化简根式，除非题目要求，否则答案应保留精确形式。
4. Completing the Square | 配方法
Completing the square is another method that transforms a quadratic into a perfect square form. It is especially useful for finding the vertex of a parabola and for solving equations when factorising is not straightforward. Start by writing x² + bx, then add and subtract (b/2)².
配方法是另一种将二次式化为完全平方式的方法。它对求抛物线的顶点以及在因式分解不直接时求解方程尤其有用。从写出 x² + bx 开始，然后加上并减去 (b/2)²。
² + 6x) - 7 = 0. Half of 6 is 3, and 3² = 9. Add and subtract 9: (x² + 6x + 9 - 9) - 7 = 0 ⇒ (x + 3)² - 16 = 0. Then (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = -7.
对于 x² + 6x - 7 = 0，改写为 (x² + 6x) - 7 = 0。6 的一半是 3，3² = 9。加减 9：(x² + 6x + 9 - 9) - 7 = 0 ⇒ (x + 3)² - 16 = 0。那么 (x + 3)² = 16，x + 3 = ±4，解得 x = 1 或 x = -7。
e 2(x² + 4x) + 5 = 0, then complete the square inside the bracket: 2[(x + 2)² - 4] + 5 = 0 ⇒ 2(x + 2)² - 8 + 5 = 0 ⇒ 2(x + 2)² = 3. Solve from there, giving x = -2 ± √3/2. This method appears in many Edexcel questions requiring exact solutions.
当 x² 的系数不是 1 时，先提取公因数。对于 2x² + 8x + 5 = 0，写成 2(x² + 4x) + 5 = 0，然后在括号内配方：2[(x + 2)² - 4] + 5 = 0 ⇒ 2(x + 2)² - 8 + 5 = 0 ⇒ 2(x + 2)² = 3。由此解得 x = -2 ± √3/2。这种方法经常出现在需要精确解的 Edexcel 试题中。
5. The Discriminant and Nature of Roots | 判别式与根的性质
here is exactly one real root (a repeated root). If D < 0, there are no real roots, meaning the parabola does not cross the x-axis.
判别式 D = b² - 4ac 告诉你二次方程有多少个实数解。如果 D > 0，有两个相异实根；如果 D = 0，恰有一个实根（重根）；如果 D < 0，无实数根，即抛物线不与 x 轴相交。
set D = 16 - 4k = 0, giving k = 4. This type of reasoning is common in higher-tier papers.
Edexcel IGCSE 考题经常要求你无需解方程而确定根的个数，或找出使二次方程有等根的常数条件。例如，要使方程 x² + 4x + k = 0 有重根，令 D = 16 - 4k = 0，得 k = 4。这类推理在进阶层试卷中很常见。
6. Graphing Quadratic Functions | 二次函数图像
. When a > 0, the parabola opens upwards (a smiling face); when a < 0, it opens downwards (a frowning face). The x-intercepts correspond to the roots of the equation, and the y-intercept is at (0, c).
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上（笑脸状）；当 a < 0 时，开口向下（哭脸状）。x 轴截距对应方程的根，y 轴截距为 (0, c)。
e vertex is (h, k). In IGCSE, you may be asked to sketch the curve, labelling intercepts and the vertex, or to solve graphical inequalities like x² - 4x + 3 > 0.
顶点（或称转折点）是曲线的极大或极小值点。配方法可以给出顶点坐标：对于 y = a(x - h)² + k，顶点为 (h, k)。在 IGCSE 中，你可能会被要求绘制曲线草图并标出截距和顶点，或者求解如图像不等式 x² - 4x + 3 > 0。
7. Solving Quadratic Inequalities | 解二次不等式
ot these on a number line and test intervals to see where the product is negative or zero.
像 x² - 3x - 10 ≤ 0 这样的二次不等式，可先求出使式子等于零的临界值。因式分解得 (x - 5)(x + 2) ≤ 0，根为 5 和 -2。在数轴上标出这两点，然后测试区间，判断乘积何处为负或零。
number line or sketch to confirm. Always consider whether the inequality includes the boundary values; in this case it does.
由于 x² 的系数为正，抛物线开口向上，式子在两根之间 ≤ 0。因此解为 -2 ≤ x ≤ 5。用数轴或草图确认。务必考虑不等式是否包含边界值；此例包含。
8. Solving Simultaneous Equations with One Quadratic | 含一个二次方程的方程组求解
x² + x - 3. Substitute the linear equation into the quadratic to get x² + x - 3 = 2x + 1.
Edexcel IGCSE 经常考查求解一个为一次、一个为二次的方程组。例如 y = 2x + 1 和 y = x² + x - 3。将一次方程代入二次式得到 x² + x - 3 = 2x + 1。
a quadratic: x² - x - 4 = 0. Solve using the quadratic formula or factorisation: x = [1 ± √1 + 16]/2 = (1 ± √17 rresponding y-values from y = 2x + 1. Give answers as exact values, often involving surds.
整理得到二次方程：x² - x - 4 = 0。使用二次公式或因式分解求解：x = [1 ± √1 + 16]/2 = (1 ± √17)/2。然后通过 y = 2x + 1 求出相应的 y 值。答案通常会包含根式，需给出精确值。
9. Word Problems Involving Quadratics | 涉及二次方程的应用题
→ x² + 5x - 36 = 0, factorise to (x + 9)(x - 4) = 0, and take the positive solution x = 4.
很多实际情境会引出二次模型，如面积问题、抛物运动和数字谜题。例如，长方形长为 (x + 5) m，宽为 x m，面积为 36 m²。列出方程 x(x + 5) = 36 → x² + 5x - 36 = 0，因式分解得 (x + 9)(x - 4) = 0，取正解 x = 4。
Always interpret the solution in the context of the problem. Negative lengths or times are usually irrelevant, so discard them. Pay close attention to units and rounding instructions if using the quadratic formula.
务必在问题情境中解释解的含义。负数长度或时间通常不合题意，应舍去。使用二次公式时要注意单位和取整指令。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
, then incorrectly deducing x = 6 or x + 5 = 6. Always rearrange to x² + 5x - 6 = 0 first.
一个常见错误是因式分解前忘记将方程设为零。学生可能试图将 x² + 5x = 6 写成 x(x + 5) = 6，然后错误地推出 x = 6 或 x + 5 = 6。一定要先化为 x² + 5x - 6 = 0。
ly with b negative. For x² - 4x - 5 = 0, a = 1, b = -4, c = -5, the formula gives x = [4 ± √16 + 20]/2. Writing b as -4 but then putting -4 in the numerator instead of +4 is a slip to watch out for.
另一个错误是在使用二次公式时符号处理不当，特别是 b 为负时。对于 x² - 4x - 5 = 0，a = 1, b = -4, c = -5，公式应为 x = [4 ± √16 + 20]/2。错误地把 b = -4 写成分子中的 -4（而非 +4）是一个需要留意的失误。
Also, when completing the square for expressions like 2x² + 8x, remember to factor out the 2 from both x terms before adding and subtracting the square term. Many students forget to multiply the subtracted constant by the leading coefficient. Double-checking each step is essential.
此外，当给诸如 2x² + 8x 的式子配方时，记得先把 2 从两个含 x 项中提出来，再加减平方项。很多学生忘记将减去的常数乘以首项系数。仔细检查每一步至关重要。
11. Revision Notes Summary Table | 复习总结表格
Here is a quick reference for the methods discussed, their key forms, and when to use them.
以下是所讨论方法的快速参考，包括关键形式及适用时机。
| Method | Key Form | Best For |
|---|---|---|
| Factorising | (px + q)(rx + s) = 0 | Simple quadratics with integer roots |
| Quadratic Formula | x = [-b ± √b² - 4ac] / (2a) | All cases, especially when factorising is hard |
| Completing the Square | a(x + p)² + q | Vertex form, exact solutions, and graph sketching |
| Discriminant | D = b² - 4ac | Nature of roots and unknown constants |
12. Practice and Exam Tips | 练习与考试技巧
In the Edexcel IGCSE exam, quadratic questions often carry multiple marks. Always show your method clearly, especially when using the formula, to earn method marks even if a small slip occurs. If a question asks for answers correct to decimal places, round only at the final step to avoid rounding errors.
在 Edexcel IGCSE 考试中，二次方程题目通常占多重分值。务必清晰地展示你的解题步骤，尤其在使用公式时，这样即使有小失误也能得到方法分。如果题目要求答案保留小数位，只在最后一步四舍五入，以避免舍入误差。
Practice a mix of problems: factorising when possible, using the formula when coefficients are large, and completing the square for vertex applications. Familiarity with the discriminant will also prepare you for those conceptual questions that ask for 'no real roots' or 'equal roots'.
练习各种类型的题目：能因式分解就因式分解，系数较大时使用公式，而对于涉及顶点的问题使用配方法。熟悉判别式也能让你为那些询问“无实根”或“等根”的概念题做好准备。
Finally, revisit word problems and simultaneous equations with quadratics, as these are often found at the end of higher-tier papers. Building speed and accuracy in these areas can significantly boost your overall grade.
最后，回顾应用题和含二次方程的方程组，因为它们经常出现在进阶层试卷的末尾。提高这些板块的速度和准确性可以显著提升你的总成绩。
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