📚 Solving Quadratic Equations | 解二次方程
completing the square, or by using the quadratic formula. Choosing the right method can save you valuable time in the exam and help you earn full marks on method marks.
二次方程几乎出现在 Edexcel IGCSE 数学的每一份试卷中，基础级和高级别都会考查。形如 ax² + bx + c = 0 的方程可以通过因式分解、配方法或求根公式来求解。在考试中选择正确的方法可以为你节省宝贵的时间，并帮助你获得完整的方法分。
1. What Is a Quadratic Equation? | 什么是二次方程？
a, b and c are constants and a ≠ 0. The highest power of x is 2, which is why it is called 'quadratic', from the Latin word 'quadratus' meaning square.
二次方程是任何可以整理成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。x 的最高次数是 2，因此被称为"二次"，源自拉丁语 'quadratus'，意为"平方"。
be zero; if a = 0, the equation becomes linear and the quadratic methods no longer apply.
例如，2x² + 5x − 3 = 0 是二次方程，而 3x + 1 = 0 是一次方程，x³ − 4x = 0 是三次方程。注意系数 a 不能为零；如果 a = 0，方程就变成一次方程，二次方程的解法就不再适用。
c crosses the x-axis. Understanding this graphical meaning helps you check whether your algebraic answers are sensible.
二次方程可以有两个不同的实根、一个重根，或者没有实根。这些根是 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。理解这一图形意义可以帮助你检查代数答案是否合理。
2. Solving by Factorisation | 因式分解法
Factorisation is usually the fastest method when the quadratic has simple integer roots. If ax² + bx + c can be written as (px + q)(rx + s), then the roots are found by setting each bracket equal to zero. This works because if the product of two expressions is zero, at least one of them must be 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。如果 ax² + bx + c 可以写成 (px + q)(rx + s)，那么令每个括号等于零即可求出根。这是因为如果两个表达式的乘积为零，那么其中至少有一个必须为零。
Example 1: Solve x² − 5x + 6 = 0.
例 1：解 x² − 5x + 6 = 0。
and add to −5. The numbers are −2 and −3. Therefore (x − 2)(x − 3) = 0, so x = 2 or x = 3.
找两个乘积为 6、和为 −5 的数。这两个数是 −2 和 −3。因此 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
Example 2: Solve 2x² + 7x + 3 = 0.
例 2：解 2x² + 7x + 3 = 0。
hat multiply to 6 and add to 7: these are 1 and 6. Split the middle term and factor by grouping:
这里的 x² 系数不是 1。将 a 和 c 相乘：2 × 3 = 6。找两个乘积为 6、和为 7 的数：它们是 1 和 6。将中间项拆分并用分组法因式分解：
2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)
expand your brackets mentally to check your factorisation before writing the final answer.
令每个括号等于零，得到 x = −½ 或 x = −3。在写出最终答案之前，务必在脑中展开括号来检查因式分解是否正确。
3. Solving by Completing the Square | 配方法
helps you find the turning point of a graph. The key identity is (x + p)² = x² + 2px + p².
配方法将二次方程改写为 (x + p)² + q 的形式。当方程不易因式分解时，这种方法特别有用，它还能帮助你找到图像的顶点。关键恒等式是 (x + p)² = x² + 2px + p²。
Example: Solve x² + 6x + 2 = 0 by completing the square.
例：用配方法解 x² + 6x + 2 = 0。
Take half of 6, which is 3, and write (x + 3)². Expanding gives x² + 6x + 9, which is 9 more than x² + 6x. Subtract 9 to compensate:
取 6 的一半，即 3，写出 (x + 3)²。展开得到 x² + 6x + 9，比 x² + 6x 多 9。减去 9 来补偿：
(x + 3)² − 9 + 2 = 0 ⟹ (x + 3)² − 7 = 0
(x + 3)² = 7 ⟹ x + 3 = ±√7 ⟹ x = −3 ± √7
Write the answer as x = −3 + √7 or x = −3 − √7 e coefficient of x² is not 1, you must first divide the whole equation by that coefficient.
答案写作 x = −3 + √7 或 x = −3 − √7 ≈ 0.646 或 −5.646。注意当 x² 的系数不是 1 时，必须先对整个方程除以该系数。
4. The Quadratic Formula | 求根公式
isation is impossible. For ax² + bx + c = 0, the roots are given by the following formula:
求根公式是一种通用方法，可以解任何二次方程，即使无法因式分解也能使用。对于 ax² + bx + c = 0，根由以下公式给出：
x = (−b ± √b² − 4ac) ÷ 2a
You will receive this formula in the Edexcel IGCSE formula booklet, but you must know how to substitute values correctly and simplify the result. Always use brackets around negative numbers when substituting.
Edexcel IGCSE 公式册中会提供此公式，但你必须知道如何正确代入数值并化简结果。代入负数时一定要使用括号。
Example: Solve 3x² − 4x − 2 = 0 using the quadratic formula.
例：用求根公式解 3x² − 4x − 2 = 0。
Here a = 3, b = −4, c = −2. Substitute carefully:
这里 a = 3，b = −4，c = −2。仔细代入：
x = (4 ± √((−4)² − 4 × 3 × (−2))) ÷ (2 × 3)
x = (4 ± √16 + 24) ÷ 6 = (4 ± √40) ÷ 6
Simplify √40 = 2√10, so x = (4 ± 2√10) ÷ 6 = (2 ± √10 ≈ 1.721 or −0.387. Leave your answer in surd form unless the question asks for a decimal.
化简 √40 = 2√10，所以 x = (4 ± 2√10) ÷ 6 = (2 ± √10) ÷ 3。即 x ≈ 1.721 或 −0.387。除非题目要求小数，否则答案保留根式形式。
5. The Discriminant | 判别式
The expression b² − 4ac inside the square root is called the discriminant. It tells you how many real roots a quadratic equation has, without solving it completely. This is a favourite area for multiple-choice and short-answer questions.
根号内的表达式 b² − 4ac 称为判别式。它告诉你二次方程有多少个实根，而无需完整解方程。这是选择题和简答题的热门考点。
- If b² − 4ac > 0, the equation has two distinct real roots, and the graph crosses the x-axis twice.
- 如果 b² − 4ac > 0，方程有两个不同的实根，图像与 x 轴有两个交点。
- has exactly one repeated root, and the graph just touches the x-axis at its turning point.
- 如果 b² − 4ac = 0，方程只有一个重根，图像在顶点处恰好与 x 轴相切。
- If b² − 4ac < 0, the equation has no real roots, and the graph never touches the x-axis.
- 如果 b² − 4ac < 0，方程没有实根，图像与 x 轴没有交点。
Example: For what values of k does the equation kx² + 4x + 1 = 0 have two distinct real roots?
例：k 为何值时，方程 kx² + 4x + 1 = 0 有两个不同的实根？
oots, the discriminant must be strictly greater than zero. Substitute a = k, b = 4, c = 1:
要有两个不同的实根，判别式必须严格大于零。代入 a = k，b = 4，c = 1：
4² − 4 × k × 1 > 0 ⟹ 16 − 4k > 0 ⟹ k < 4
Since a ≠ 0, we must also have k ≠ 0. The final answer is k < 4 and k ≠ 0. Be careful not to forget this extra condition.
由于 a ≠ 0，还必须有 k ≠ 0。最终答案是 k < 4 且 k ≠ 0。注意不要忘记这个额外条件。
6. Choosing the Best Method | 选择最佳方法
Exam questions often do not tell you which method to use. Choosing efficiently can reduce errors and save time. Here is a simple decision guide based on the form of the equation.
考试题目通常不会告诉你使用哪种方法。高效的选择可以减少错误并节省时间。以下是根据方程形式给出的简单选择指南。
| Equation Form | Best Method | Reason |
| x² + bx + c = 0 with simple integer roots | Factorisation | Fastest and uses fewest steps |
| x² + 2px form, or need turning point | Completing the square | Reveals vertex coordinates directly |
| ax² + bx + c = 0 where a ≠ 1 and factorisation is hard | Quadratic formula | Always works, guaranteed reliable |
Always check your answers by substituting back into the original equation. In mark schemes, a correct substitution check can earn you a method mark even if your factorisation went wrong.
始终通过将答案代回原方程来检查。在评分标准中，即使因式分解出错，正确的代入验算也能为你赢得方法分。
7. Quadratic Word Problems | 二次方程应用题
Edexcel IGCSE examiners love to set quadratics in real-world contexts, such as areas, projectile motion and number puzzles. The key skill is translating the words into an equation and then solving it using your preferred method.
Edexcel IGCSE 考官喜欢将二次方程放在实际情境中考查，例如面积、抛体运动和数字谜题。关键技能是将文字转化为方程，然后用你偏好的方法求解。
Example: A rectangle has length (x + 5) cm and width (x − 2) cm. Its area is 54 cm². Find the value of x.
例：一个长方形的长为 (x + 5) cm，宽为 (x − 2) cm，面积为 54 cm²。求 x 的值。
Set up the equation using area = length × width:
利用面积 = 长 × 宽 建立方程：
(x + 5)(x − 2) = 54 ⟹ x² + 3x − 10 = 54 ⟹ x² + 3x − 64 = 0
This does not factorise, so use the quadratic formula with a = 1, b = 3, c = −64:
此式无法因式分解，因此用求根公式，其中 a = 1，b = 3，c = −64：
x = (−3 ± √9 + 256) ÷ 2 = (−3 ± √265) ÷ 2
6.64 cm. Always reject negative solutions in geometric contexts and state your reasoning.
所以 x ≈ 6.64 或 x ≈ −9.64。由于长度不能为负，舍去负根。因此 x ≈ 6.64 cm。在几何情境中一定要舍去负数解并说明理由。
8. Quadratic Graphs and Roots | 二次函数图像与根
wo roots, the graph crosses the axis twice; if one repeated root, it touches once; if no roots, it never touches.
二次方程的根对应图像 y = ax² + bx + c 与 x 轴的交点。如果方程有两个根，图像与轴相交两次；如果有一个重根，则相切一次；如果没有根，则永不相交。
f a > 0, the graph is a U-shape and the vertex is a minimum; if a < 0, the graph is an n-shape and the vertex is a maximum.
图像的顶点可以通过配方法形式 y = (x + p)² + q 求出。顶点坐标为 (−p, q)。如果 a > 0，图像是 U 形，顶点为最小值；如果 a < 0，图像是 ∩ 形，顶点为最大值。
Example: y = x² − 8x + 15 in completed square form and state the coordinates of the turning point.
例：将 y = x² − 8x + 15 写成配方法形式，并写出顶点坐标。
y = (x − 4)² − 16 + 15 = (x − 4)² − 1
= 5, which confirms the graph crosses the x-axis halfway between the vertex and each root.
因此顶点为 (4, −1)。由于 x² 的系数为正，这是最小值点。方程 x² − 8x + 15 = 0 的根为 x = 3 和 x = 5，这证实图像与 x 轴交点的横坐标对称分布在顶点两侧。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks on quadratics for avoidable reasons. Knowing these common pitfalls can boost your grade quickly. Here are the most frequent errors seen in Edexcel IGCSE examiners' reports.
许多学生因为可以避免的原因在二次方程题目上失分。了解这些常见陷阱可以快速提高你的成绩。以下是 Edexcel IGCSE 考官报告中反复提到的常见错误。
- c = 0 before factorising or applying the formula. Always move all terms to one side first.
- 忘记将方程整理成 ax² + bx + c = 0 的形式就进行因式分解或套用公式。务必先将所有项移到一边。
- Sign errors when substituting negative values into the quadratic formula. Write (−b)² correctly and use brackets throughout.
- 将负值代入求根公式时出现符号错误。正确写出 (−b)²，并全程使用括号。
- Dropping the ± symbol and losing one root. The expression x = (−b ± √...) ÷ 2a always gives two answers when the discriminant is positive.
- 漏掉 ± 符号而丢失一个根。当判别式为正时，x = (−b ± √...) ÷ 2a 总是给出两个答案。
- Stopping after factorisation without actually solving for x. Setting each bracket to zero is required to earn full marks.
- 因式分解后没有继续解出 x。必须令每个括号等于零才能获得满分。
In the exam, show every step clearly. Edexcel mark schemes award method marks for correct factorisation, completing the square, or substitution into the formula, even if your final answer is wrong. A blank solution earns nothing.
考试中要清晰地写出每一步。Edexcel 评分标准会对正确的因式分解、配方或代入公式给予方法分，即使最终答案错误。空白解答则零分。
10. Practice Questions | 练习题目
Try these questions in exam conditions, then check your answers below. Aim to complete each question in under three minutes.
请在模拟考试条件下尝试以下题目，然后对照下面的答案。争取每题在三分钟内完成。
Q1. Solve x² − 4x − 21 = 0 by factorisation.
题 1. 用因式分解法解 x² − 4x − 21 = 0。
Q2. Solve 2x² + 8x + 3 = 0 by completing the square. Leave your answer in surd form.
题 2. 用配方法解 2x² + 8x + 3 = 0，答案保留根式形式。
Q3. Find the value of the discriminant of 5x² − 2x + 1 and state the number of real roots.
题 3. 求 5x² − 2x + 1 的判别式值，并说明实根的个数。
Q4. The sum of a number and its reciprocal is 4. Form a quadratic equation and solve it, giving answers correct to 2 decimal places.
题 4. 一个数与其倒数的和为 4。建立二次方程并求解，答案精确到 2 位小数。
Answers: Q1: x = 7 or x = −3. Q2: x = −2 ± (√10 r −3.58. Q3: discriminant = (−2)² − 4 × 5 × 1 = −16 < 0, so no real roots. Q4: x + 1/x = 4 ⟹ x² − 4x + 1 = 0 ⟹ x ≈ 3.73 or 0.27.
答案：题 1：x = 7 或 x = −3。题 2：x = −2 ± (√10)/2 ≈ −0.42 或 −3.58。题 3：判别式 = (−2)² − 4 × 5 × 1 = −16 < 0，所以没有实根。题 4：x + 1/x = 4 ⟹ x² − 4x + 1 = 0 ⟹ x ≈ 3.73 或 0.27。
Mastering quadratics takes practice, but the reward is significant: this topic appears in multiple questions across Paper 1 and Paper 2, often worth 4 to 6 marks each. Work through past paper questions on quadratics until the methods become automatic.
掌握二次方程需要练习，但回报丰厚：这一主题在 Paper 1 和 Paper 2 的多道题目中出现，每道通常占 4 到 6 分。请反复练习历年真题中的二次方程题目，直到这些方法变得得心应手。
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