📚 Mastering Quadratic Equations | 二次方程完全攻略
Quadratic equations are one of the most heavily tested topics in the Edexcel IGCSE Mathematics syllabus. You will meet them in both Paper 1 and Paper 2, often combined with graphs, geometry, sequences and word problems. This article covers every essential skill, from expanding brackets to applying the quadratic formula, with worked examples and exam-style advice.
二次方程是 Edexcel IGCSE 数学考纲中考查频率最高的内容之一。卷一和卷二都会出现，并经常与图象、几何、数列及应用题相结合。本文覆盖从展开括号到套用求根公式的全部核心技能，配有例题和考场建议。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is exactly 2.
二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数恰好是 2。
For example, x² − 4x + 3 = 0 and 2x² + 5x − 1 = 0 are quadratic equations. However, an equation such as x³ + 2x = 0 is not quadratic, because the highest power is 3.
例如，x² − 4x + 3 = 0 和 2x² + 5x − 1 = 0 都是二次方程。但 x³ + 2x = 0 不是二次方程，因为最高次数是 3。
If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential. In the Edexcel IGCSE exam, you must be able to recognise quadratics in any form, including those with fractions or decimals.
如果 a = 0，方程就退化为一次方程，因此 a ≠ 0 这一条件必不可少。在 Edexcel IGCSE 考试中，你必须能够识别任何形式的二次方程，包括含有分数或小数的情形。
2. Expanding Double Brackets | 展开二项括号
Before solving or factorising, you must be fluent in expanding double brackets. Use the rule: multiply each term in the first bracket by each term in the second bracket.
在求解或因式分解之前，你必须熟练展开二项括号。规则是：将第一个括号中的每一项乘以第二个括号中的每一项。
For example:
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
The general pattern is (x + p)(x + q) = x² + (p + q)x + pq. The coefficient of x is the sum p + q, and the constant term is the product pq.
一般规律是 (x + p)(x + q) = x² + (p + q)x + pq。x 的系数是 p + q，常数项是 pq。
When the coefficient a is not 1, for example (2x + 1)(3x − 4), multiply each term carefully:
当 a 的系数不是 1 时，例如 (2x + 1)(3x − 4)，要仔细逐项相乘：
(2x + 1)(3x − 4) = 6x² − 8x + 3x − 4 = 6x² − 5x − 4
Always simplify like terms after expanding. This skill is also needed when manipulating the numerator in the quadratic formula.
展开后务必合并同类项。这一技能在化简求根公式中的分子时同样需要。
3. Factorising Quadratics | 因式分解二次式
Factorising is the reverse of expanding. For x² + 8x + 15, look for two numbers that multiply to 15 and add to 8. Those numbers are 3 and 5.
因式分解是展开的逆运算。对于 x² + 8x + 15，寻找两个数相乘得 15、相加得 8。这两个数是 3 和 5。
x² + 8x + 15 = (x + 3)(x + 5)
For quadratics with a ≠ 1, such as 2x² + 7x + 3, use the ac method: multiply a and c to get 6, then find two factors of 6 that add to 7. These are 1 and 6.
对于 a ≠ 1 的二次式，例如 2x² + 7x + 3，使用 ac 法：先算 a × c 得 6，再找两个 6 的因数，使它们相加为 7。这两个数是 1 和 6。
Rewrite the middle term and factorise by grouping:
改写中间项，然后分组因式分解：
2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)
Always check your answer by expanding again. Most Edexcel IGCSE factorising questions are worth 2–3 marks and require fully factorised answers.
完成后务必重新展开验证。多数 Edexcel IGCSE 的因式分解题分值为 2–3 分，并要求写出完整分解后的结果。
4. Solving by Factorising | 用因式分解法求解
The key principle is the zero product property: if A × B = 0, then A = 0 or B = 0. This allows us to split a quadratic equation into two simple linear equations.
关键原理是零积性质：如果 A × B = 0，那么 A = 0 或 B = 0。这使我们能把二次方程拆分成两个简单的一次方程。
Example: solve x² − 2x − 15 = 0.
例：解 x² − 2x − 15 = 0。
First factorise: (x − 5)(x + 3) = 0. Then set each bracket to zero:
先因式分解：(x − 5)(x + 3) = 0。再令每个括号分别为零：
x − 5 = 0 or x + 3 = 0, so x = 5 or x = −3
Always rearrange the equation into the form ax² + bx + c = 0 before factorising. For example, x² = 6x − 8 becomes x² − 6x + 8 = 0, which factorises as (x − 2)(x − 4) = 0.
因式分解前必须把方程整理成 ax² + bx + c = 0 的形式。例如 x² = 6x − 8 应变为 x² − 6x + 8 = 0，分解为 (x − 2)(x − 4) = 0。
If a quadratic is a difference of two squares, use the special rule a² − b² = (a − b)(a + b). For example, x² − 9 = (x − 3)(x + 3).
如果二次式是平方差，使用特殊公式 a² − b² = (a − b)(a + b)。例如 x² − 9 = (x − 3)(x + 3)。
5. Completing the Square | 配方法
Completing the square rewrites x² + bx + c in the form (x + p)² + q. Take half of the coefficient of x to find p.
配方法将 x² + bx + c 改写为 (x + p)² + q 的形式。取 x 系数的一半即可得到 p。
Example: x² + 6x + 2. Half of 6 is 3, so we write (x + 3)². Because (x + 3)² = x² + 6x + 9, we must correct:
例：x² + 6x + 2。6 的一半是 3，于是写成 (x + 3)²。因为 (x + 3)² = x² + 6x + 9，所以需要修正：
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
To solve a quadratic by completing the square, isolate the squared term and then take the square root of both sides, remembering the ± sign.
用配方法解二次方程时，先孤立平方项，然后对方程两边同时开平方，注意加上 ± 号。
Example: solve x² + 6x + 2 = 0.
例：解 x² + 6x + 2 = 0。
(x + 3)² − 7 = 0 → (x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7
Completing the square also gives the turning point of a quadratic graph, as explained in Section 9.
配方法还能直接给出二次函数图象的顶点坐标，详见第 9 节。
6. The Quadratic Formula | 二次求根公式
When factorising is difficult or impossible, use the quadratic formula. For any quadratic ax² + bx + c = 0:
当因式分解困难或无法分解时，使用二次求根公式。对于任意二次方程 ax² + bx + c = 0：
x = (−b ± √(b² − 4ac)) / 2a
This formula is provided on the Edexcel IGCSE formula sheet, but you must know how to substitute correctly and simplify the result. Always write the equation in standard form first.
该公式会印在 Edexcel IGCSE 公式表上，但你必须会正确代入并化简结果。一定要先把方程写成标准形式。
Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3.
例：解 2x² − 4x − 3 = 0。其中 a = 2，b = −4，c = −3。
x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2
Give your answer as an exact surd, or to one or two decimal places if the question asks for a decimal approximation. Edexcel IGCSE often requires exact surd answers in the final part of a question.
答案可以保留根号精确值，如果题目要求近似值，则保留一位或两位小数。Edexcel IGCSE 常在题目的最后一部分要求写出精确的根式答案。
7. The Discriminant | 判别式
The discriminant is the value under the square root in the quadratic formula: Δ = b² − 4ac. It tells us how many real roots a quadratic equation has.
判别式是求根公式中根号内的部分：Δ = b² − 4ac。它告诉我们二次方程有多少个实数根。
| 判别式的值 | 根的个数与类型 |
| Δ > 0 | 两个不同的实数根 |
| Δ = 0 | 一个重根（两个相等实数根） |
| Δ < 0 | 没有实数根 |
Table: The discriminant determines the number of real roots. | 表格：判别式决定实数根的个数。
For example, x² − 4x + 4 = 0 has Δ = 16 − 16 = 0, so it has one repeated root, x = 2. The graph touches the x-axis at exactly one point.
例如 x² − 4x + 4 = 0 的 Δ = 16 − 16 = 0，所以它有一个重根 x = 2。图象与 x 轴恰好相切于一点。
In exam problems, you may be asked to find the range of k such that a quadratic has two real roots. Write the inequality b² − 4ac > 0 and solve it carefully.
考试中常要求求出 k 的范围使二次方程有两个实数根。写出不等式 b² − 4ac > 0 并仔细求解。
8. Graphs of Quadratic Functions | 二次函数图象
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards like a smile; if a < 0, it opens downwards like a frown.
y = ax² + bx + c 的图象是抛物线。若 a > 0，抛物线开口向上，形如微笑；若 a < 0，开口向下，形如皱眉。
The y-intercept is always (0, c), because substituting x = 0 leaves y = c. The x-intercepts are the roots of the equation ax² + bx + c = 0.
y 截距始终是 (0, c)，因为代入 x = 0 后只剩下 y = c。x 截距就是方程 ax² + bx + c = 0 的根。
When sketching, show the desired features clearly: the y-intercept, the x-intercepts (if any) and the turning point. Draw a smooth curve using a ruler for the axes and freehand for the curve.
画草图时要清楚标出以下特征：y 截距、x 截距（如果存在）以及顶点坐标。用直尺画坐标轴，曲线则徒手画平滑。
If the discriminant is negative, the parabola does not cross the x-axis at all. Use this to check whether your sketch is sensible.
如果判别式为负，抛物线完全不与 x 轴相交。可以利用这一点检查你的草图是否合理。
9. Finding the Turning Point | 求顶点坐标
The turning point is the maximum or minimum point of the parabola. Completing the square gives it directly: if y = a(x + p)² + q, the vertex is (−p, q).
顶点是抛物线的最高点或最低点。配方法可以直接给出顶点：若 y = a(x + p)² + q，则顶点为 (−p, q)。
Example: y = (x − 2)² + 5 has vertex at (2, 5). Because the coefficient of the squared term is positive, this is a minimum point.
例：y = (x − 2)² + 5 的顶点是 (2, 5)。由于平方项的系数为正，这是个最小值点。
Alternatively, the axis of symmetry is x = −b / 2a. Substitute this x-value back into the equation to find the y-coordinate of the vertex.
另一种方法是利用对称轴 x = −b / 2a。把该 x 值代回方程，即可求得顶点的 y 坐标。
For y = x² − 4x + 1, we have x = 4 / 2 = 2, and y = 4 − 8 + 1 = −3. So the vertex is (2, −3).
对 y = x² − 4x + 1，x = 4 / 2 = 2，y = 4 − 8 + 1 = −3。所以顶点是 (2, −3)。
The line of symmetry always passes through the vertex, so once you know one half of the parabola you can reflect to draw the other half.
对称轴一定经过顶点，因此画出一半抛物线后，可以通过轴对称反射画出另一半。
10. Real-World Applications | 实际应用
Quadratics model many real-life situations. A classic IGCSE problem involves finding the length and width of a rectangle from its area and perimeter.
二次方程可以建模许多实际情境。一个经典的 IGCSE 题目是已知面积和周长求矩形的长和宽。
Example: a rectangular garden has length (x + 4) m and width (x − 1) m, and its area is 28 m². Then (x + 4)(x − 1) = 28, which expands to x² + 3x − 4 = 28, giving x² + 3x − 32 = 0.
例：一块长方形花坛的长为 (x + 4) m，宽为 (x − 1) m，面积为 28 m²。则 (x + 4)(x − 1) = 28，展开得 x² + 3x − 4 = 28，即 x² + 3x − 32 = 0。
Projectile motion is another common context. The height h metres of a ball after t seconds might be h = 5t − 4.9t². Solving h = 0 gives the time when the ball hits the ground.
抛体运动是另一个常见背景。球在 t 秒后的高度 h 米可能满足 h = 5t − 4.9t²。令 h = 0 求解，即可得到球落地的时间。
When solving applied problems, check that your answer makes sense in the context. A length or time cannot be negative, so reject any impossible root.
解答应用题时，要检查答案是否符合实际背景。长度或时间不能为负数，所以必须舍去不合理的根。
11. Common Exam Mistakes | 常见考场错误
Many marks are lost each year due to avoidable errors. The table below lists the most frequent pitfalls in Edexcel IGCSE quadratic questions.
每年都有大量分数因可避免的错误而丢失。下表列出 Edexcel IGCSE 二次方程题目中最常见的陷阱。
| 常见错误 | 正确做法 |
| 两边同时除以 x，丢失一根 | 先移项再因式分解 |
| 开平方时忘记 ± | x² = k 时写 x = ±√k |
| 公式中 a 或 c 的符号代错 | 代入前先整理出标准形式 |
| 因式分解后括号内符号写反 | 展开检查一次 |
Always write down intermediate steps. In a two-mark question, even if your final answer is wrong, you may still earn a method mark for a correct equation or factorisation.
始终写出中间步骤。在一道 2 分题中，即使最终答案错误，只要方程或因式分解正确，仍可能获得方法分。
Finally, give your answer in the exact form required. If the question says ‘give your answer correct to 2 decimal places’, do not leave a surd.
最后，按题目要求的形式作答。如果题目要求”答案保留两位小数”，就不要保留根号。
12. Quick Revision Checklist | 考前快速清单
Use this checklist before your exam to confirm you have mastered each skill.
考试前使用这份清单，确认自己已经掌握每一项技能。
- I can expand and simplify double brackets correctly. | 我能正确展开并化简二项括号。
- I can factorise quadratics with a = 1 and a ≠ 1. | 我能分解 a = 1 和 a ≠ 1 的二次式。
- I can solve quadratics by factorising. | 我能用因式分解法解二次方程。
- I can complete the square and solve using it. | 我会配方法并会用它求解方程。
- I can use the quadratic formula with correct substitution. | 我能正确代入并使用二次求根公式。
- I can interpret the discriminant and state the number of roots. | 我能解释判别式并判断根的个数。
- I can sketch quadratic graphs labelling intercepts and the vertex. | 我能画二次函数
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