📚 Solving Quadratic Equations: A Complete Guide | 一元二次方程的求解方法详解
A quadratic equation is one of the most fundamental topics in A-Level mathematics. It appears in virtually every module — from pure mathematics to mechanics and statistics — and mastering its solution methods is essential for exam success. This guide provides a thorough, step-by-step breakdown of every method you need to know for the Edexcel syllabus.
一元二次方程是 A-Level 数学中最基础也最重要的知识点之一。无论纯数、力学还是统计学模块，它几乎无处不在。掌握其求解方法，是考试取得高分的关键。本文将针对 Edexcel 考纲，逐一详解你必须掌握的各种解法。
1. Standard Form and Basic Concepts | 标准形式与基本概念
A quadratic equation in one variable is any equation that can be written in the standard form:
一元二次方程是指可以写成以下标准形式的含一个未知数的方程：
ax² + bx + c = 0, 其中 a ≠ 0
Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. A root (or solution) of the equation is any value of x that satisfies the equation. The degree of the equation is 2, which means a quadratic equation can have at most two distinct real roots.
其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。方程的根（或解）是满足方程的 x 的取值。该方程为二次方程，因此最多可以有两个不同的实数根。
e standard form ax² + bx + c = 0. This is a crucial first step that many students overlook — especially when the equation is given in a disguised form such as x² = 5x − 6.
在解任何一元二次方程之前，务必先将其化为标准形式 ax² + bx + c = 0。这是许多学生容易忽略的关键第一步——特别是当方程以隐含形式给出时，例如 x² = 5x − 6。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when the quadratic expression factorises neatly. The principle is based on the zero product property: if the product of two expressions is zero, then at least one of the expressions must be zero.
因式分解法通常是在二次表达式能够整齐分解时最快捷的方法。其原理基于零乘积性质：若两个表达式的乘积为零，则至少有一个表达式必须为零。
若 pq = 0，则 p = 0 或 q = 0
Case 1: When a = 1 (monic quadratic). seek two numbers whose product is 6 and whose sum is 5. These numbers are 2 and 3. Hence:
情况一：a = 1（首项系数为 1）。 例如 x² + 5x + 6 = 0。我们需要找两个数，它们的乘积为 6，和为 5。这两个数是 2 和 3。因此：
x² + 5x + 6 = (x + 2)(x + 3) = 0
Applying the zero product property gives x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.
由零乘积性质可得 x + 2 = 0 或 x + 3 = 0，因此 x = −2 或 x = −3。
Case 2: When a ≠ 1 (non-monic quadratic). For example, 2x² + 7x + 3 = 0. Multiply a and c o numbers whose product is 6 and whose sum is 7: these are 6 and 1. Split the middle term:
情况二：a ≠ 1（首项系数不为 1）。 例如 2x² + 7x + 3 = 0。将 a 与 c 相乘：2 × 3 = 6。找两个数使其乘积为 6、和为 7：即 6 和 1。拆分中间项：
2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0
Thus x = −½ or x = −3.
因此 x = −½ 或 x = −3。
Special case: Difference of two squares. ) = 0, giving x = 3 or x = −3. This pattern a² − b² = (a − b)(a + b) is tested frequently.
特殊情形：平方差公式。 对于 x² − 9 = 0，可写成 (x − 3)(x + 3) = 0，得到 x = 3 或 x = −3。公式 a² − b² = (a − b)(a + b) 在考试中出现频率很高。
3. Solving by Completing the Square | 配方法
Completing the square transforms a quadratic into the form (x + p)² + q, which makes the roots immediately visible and also helps locate the vertex of the parabola. This method always works, even when the expression does not factorise nicely.
配方法将二次式转化为 (x + p)² + q 的形式，使根一目了然，也有助于确定抛物线的顶点。该方法始终适用，即使表达式无法整齐因式分解。
Step 1: Ensure the coefficient of x² is 1. If it is not, divide the entire equation by a first.
第一步：确保 x² 的系数为 1。若非 1，先将整个方程除以 a。
Step 2: Halve the coefficient of x, square it, and add and subtract this value.
第二步：取 x 系数的一半，平方后加上再减去该值。
Example: Solve x² + 6x + 2 = 0.
示例：解方程 x² + 6x + 2 = 0。
The coefficient of x is 6. Half of 6 is 3, and 3² = 9. We add and subtract 9:
x 的系数为 6。6 的一半为 3，3² = 9。我们加上再减去 9：
x² + 6x + 9 − 9 + 2 = 0 → (x + 3)² − 7 = 0
Now rearrange:
接着整理：
(x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7
This gives the two roots x = −3 + √7 and x = −3 − √7. Note that when the right-hand side is positive, there are two distinct real roots; if it is zero, there is exactly one repeated root; if negative, there are no real roots.
由此得到两个根 x = −3 + √7 和 x = −3 − √7。注意：当等号右侧为正时，有两个不同的实数根；为零时，恰好有一个重根；为负时，则没有实数根。
e vertex is located at (h, k). In the example above, y = (x + 3)² − 7 has vertex (−3, −7).
顶点式在坐标几何中特别有用。对于写作 a(x − h)² + k 的二次函数，顶点位于 (h, k)。上例中 y = (x + 3)² − 7 的顶点为 (−3, −7)。
4. Solving by the Quadratic Formula | 求根公式法
The quadratic formula is the most universal method — rdless of whether the expression factorises. For ax² + bx + c = 0, the roots are given by:
求根公式是最通用的方法——它能解任意一元二次方程，无论表达式是否能因式分解。对于 ax² + bx + c = 0，根由下式给出：
x = (−b ± √b² − 4ac) / 2a
This formula is derived by completing the square on the general form. You must memorise it — it is not provided in the Edexcel formula booklet for the pure mathematics papers at A-Level.
该公式通过对一般形式配方推导得出。你必须牢记它——Edexcel 纯数试卷的公式册中并不会提供此公式。
Worked example: Solve 3x² − 5x − 2 = 0 using the quadratic formula.
例题：用求根公式解 3x² − 5x − 2 = 0。
Here a = 3, b = −5, c = −2. Substitute into the formula:
此处 a = 3，b = −5，c = −2。代入公式：
x = (5 ± √((−5)² − 4 × 3 × (−2))) / (2 × 3) = (5 ± √25 + 24) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6
So x = (5 + 7) / 6 = 2, or x = (5 − 7) / 6 = −⅓. The two roots are x = 2 and x = −⅓.
因此 x = (5 + 7) / 6 = 2，或 x = (5 − 7) / 6 = −⅓。两根为 x = 2 和 x = −⅓。
In Edexcel exams, the quadratic formula typically appears in non-calculator papers. Practising arithmetic with fractions and surds is essential to avoid careless errors.
在 Edexcel 考试中，求根公式通常出现在非计算器试卷中。熟练进行分数和根式的运算至关重要，以避免粗心失误。
5. Solving by Graphing | 图像法
s method rarely gives exact answers but provides valuable insight into the nature of the roots.
从图像上看，一元二次方程 ax² + bx + c = 0 的根对应抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。这种方法很少给出精确答案，但能帮助你直观理解根的性质。
Key features of the graph to identify:
需要识别图像的关键特征：
- Shape: If a > 0, the parabola opens upward ("smiley face"); if a < 0, it opens downward ("frown").
- 形状：若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。
- y-intercept: The graph crosses the y-axis at (0, c).
- y 轴截距：图像在 (0, c) 处与 y 轴相交。
- x-intercepts: These are precisely the roots of the equation.
- x 轴截距：这些正是方程的根。
- Vertex: The turning point is at x = −b / 2a; substitute back to find the y-coordinate.
- 顶点：转折点在 x = −b / 2a 处；代回原式可得 y 坐标。
In exam questions, you may be asked to sketch the graph of a quadratic and use it to determine the number of real roots. The number of x-intercepts tells you immediately: two intercepts means two distinct real roots, one intercept means a repeated root, and no intercept means no real roots.
在考试中，你可能会被要求画出二次函数图像并据此判断实数根的个数。x 轴交点的数量直接告诉你答案：两个交点对应两个不同的实数根，一个交点对应一个重根，没有交点则没有实数根。
6. The Discriminant | 判别式
The discriminant, denoted Δ (delta), is the expression under the square root in the quadratic formula:
判别式（记作 Δ）是求根公式中根号内的表达式：
Δ = b² − 4ac
The sign of the discriminant determines the nature of the roots:
判别式的符号决定了根的性质：
| Value of Δ | Δ 的值 | Nature of Roots | 根的性质 | Graph | 图像 |
|---|---|---|
| Δ > 0 | Two distinct real roots; if Δ is a perfect square, the equation also factorises | Crosses x-axis at two points |
| Δ = 0 | One repeated real root (two equal roots) | Touches x-axis at one point (tangent) |
| Δ < 0 | No real roots — two complex conjugate roots | Does not intersect x-axis at all |
al roots; if Δ = 0, there is exactly one repeated root; if Δ < 0, there are no real roots.
中文对照：若 Δ > 0，有两个不同的实数根；若 Δ = 0，恰好有一个重根；若 Δ < 0，则没有实数根。
Edexcel-style question: Find the range of values of k for which the equation x² + kx + 9 = 0 has two distinct real roots.
Edexcel 风格题型：求使方程 x² + kx + 9 = 0 有两个不同的实数根时 k 的取值范围。
For two distinct real roots, we require Δ > 0:
要有两个不同的实数根，需要 Δ > 0：
k² − 4 × 1 × 9 > 0 → k² − 36 > 0 → (k − 6)(k + 6) > 0
Solving this inequality gives k < −6 or k > 6.
解此不等式得 k < −6 或 k > 6。
This type of question combines quadratic equations with inequality skills — a common crossover topic in Edexcel Paper 1.
这类题目将一元二次方程与不等式技能结合——是 Edexcel 卷一中的常见跨知识点题型。
7. Choosing the Best Method | 如何选择最优解法
Knowing when to use each method saves time and reduces errors. The table below summarises the recommended approach:
知道何时使用哪种方法可以节省时间并减少失误。下表总结了推荐的策略：
| Situation | 情形 | Recommended Method | 推荐方法 |
|---|---|
| Expression factorises easily; Δ is a perfect square | Factorisation (quickest) |
| Expression does not factorise; exact surd answers needed | Quadratic formula |
| Need vertex form or graph sketching; equation has no x-term | Completing the square |
| Approximate solutions acceptable; checking results | Graphing / calculator |
| Question asks for nature of roots or parameter range | Discriminant analysis |
In Chinese: 能因式分解时优先用因式分解法；不能分解且需要精确根式答案时用求根公式；需要顶点式或画图时用配方法；只需近似解时用图像法或计算器；题目问根的性质或参数范围时，使用判别式分析。
A useful exam strategy: after finding roots by any method, substitute them back into the original equation to verify. This takes only seconds and catches most arithmetic mistakes.
一个实用的考试策略：无论用哪种方法求出根之后，代回原方程验证一下。这只需几秒钟，却能发现绝大多数运算错误。
8. Word Problems and Applications | 应用题与实际问题
In Edexcel A-Level, quadratic equations frequently appear in contextual problems. A classic example involves projectile motion: the height h (in metres) of a ball thrown upward at time t seconds is given by h = 20t − 5t². To find when the ball hits the ground, set h = 0:
在 Edexcel A-Level 中，一元二次方程经常出现在实际情境题中。一个经典例子是抛体运动：球被向上抛出后，其在 t 秒时的高度 h（米）满足 h = 20t − 5t²。求球何时落地，只需令 h = 0：
20t − 5t² = 0 → 5t(4 − t) = 0 → t = 0 或 t = 4
when the ball hits the ground. Always interpret both roots in the context of the problem.
解 t = 0 代表抛出瞬间，t = 4 秒代表落地时刻。务必结合题意解释每个根的实际意义。
3x − 40 = 0. Factorising: (x + 8)(x − 5) = 0. Since a width cannot be negative, x = 5 cm.
另一个常见应用是几何问题。例如，矩形的面积为 40 cm²，长比宽多 3 cm。若宽为 x cm，则 x(x + 3) = 40，展开得 x² + 3x − 40 = 0。因式分解：(x + 8)(x − 5) = 0。宽度不能为负，因此 x = 5 cm。
When setting up equations from word problems, clearly define your variable, translate the conditions into algebra, and check that your final answer makes sense in the original context.
从文字题建模时，要明确定义未知量，将条件转化为代数式，并检查最终答案是否符合原题情境。
9. Common Mistakes and Exam Tips | 常见错误与应试技巧
Here are the most frequent pitfalls Edexcel candidates encounter, along with practical advice:
以下列举 Edexcel 考生最常踩的坑以及实用建议：
- Not rearranging to standard form first. Always write the equation as ax² + bx + c = 0 before applying any method.
- 未先将方程化为标准形式。 在使用任何方法之前，务必先将方程写成 ax² + bx + c = 0。
- Losing the ± sign when taking square roots. From x² = 9, it is x = ±3, not just x = 3.
- 取平方根时遗漏 ± 号。 由 x² = 9 应得 x = ±3，而不仅仅是 x = 3。
- Dividing both sides by x (or by an expression containing x). can eliminate a root. For x² = 5x, do not divide by x; instead factorise as x(x − 5) = 0.
- 两边同除以 x（或含 x 的表达式）。 这可能会丢失一个根。对于 x² = 5x，切勿两边除以 x，而应因式分解为 x(x − 5) = 0。
- Arithmetic slips in the quadratic formula. Write down the values of a, b, c explicitly before substituting.
- 在求根公式中出现运算失误。 先明确写出 a、b、c 的值，再代入计算。
- Misinterpreting "no real roots." If Δ < 0, do not write "no solution" — the equation has complex roots, which you may be asked to find in later pure mathematics topics.
- 误读"无实数根"的含义。 若 Δ < 0，不要写"无解"——方程仍有复数根，后续纯数内容中你可能会被要求求解。
- Forgetting to check whether a = 0. If a = 0, the equation is linear, not quadratic, and the quadratic formula does not apply.
- 忘记检查 a 是否为 0。 若 a = 0，方程为一次方程而非二次方程，求根公式不适用。
In the Edexcel exam, always show your working clearly. In "show that" questions, every step of algebra must be visible to earn method marks, even if your final answer is correct.
在 Edexcel 考试中，务必清晰展示计算过程。在"证明"类题目中，即使最终答案正确，每一步代数变形也必须完整写出才能获得方法分。
10. Practice Questions | 巩固练习
Work through the following questions to consolidate your understanding. Answers and brief hints are provided below.
请完成以下练习以巩固理解。参考答案与提示附在下方。
Q1. Solve x² − 7x + 12 = 0 by factorisation.
练习 1：用因式分解法解 x² − 7x + 12 = 0。
Q2. Solve 2x² + 3x − 2 = 0 by any method.
练习 2：用任意方法解 2x² + 3x − 2 = 0。
Q3. Use completing the square to solve x² − 4x − 1 = 0, leaving your answers in surd form.
练习 3：用配方法解 x² − 4x − 1 = 0，答案保留根式形式。
Q4. Determine the nature of the roots of 4x² − 12x + 9 = 0 without solving.
练习 4：不解方程，判断 4x² − 12x + 9 = 0 的根的性质。
Q5. Find the values of k such that the equation kx² + 6x + 1 = 0 has no real roots.
练习 5：求使方程 kx² + 6x + 1 = 0 无实数根的 k 的取值范围。
Answers and hints:
答案与提示：
A1: (x − 3)(x − 4) = 0, so x = 3 or x = 4.
答 1：(x − 3)(x − 4) = 0，因此 x = 3 或 x = 4。
A2: (2x − 1)(x + 2) = 0, so x = ½ or x = −2.
答 2：(2x − 1)(x + 2) = 0，因此 x = ½ 或 x = −2。
A3: (x − 2)² − 5 = 0 → x = 2 ± √5.
答 3：(x − 2)² − 5 = 0 → x = 2 ± √5。
A4: Δ = (−12)² − 4 × 4 × 9 = 144 − 144 = 0, so there is one repeated real root.
答 4：Δ = (−12)² − 4 × 4 × 9 = 144 − 144 = 0，因此有一个重根。
A5: iscriminant condition must be re-examined. The strict requirement for no real roots with a quadratic is Δ < 0, so k > 9.
答 5：Δ = 36 − 4k < 0 → k > 9。注意若 k = 0，方程退化为一次方程，不符合题意；对于二次方程无实数根的严格条件为 Δ < 0，因此 k > 9。
Mastering quadratic equations is not just about memorising the quadratic formula — it is about understanding why each method works, when to apply it, and how to verify your answers. With consistent practice, these techniques will become second nature, giving you a solid foundation for more advanced topics such as inequalities, functions, and calculus. Keep practising, and good luck with your Edexcel examinations!
掌握一元二次方程不仅仅是背会求根公式——更重要的是理解每种方法背后的原理、何时使用它们以及如何验证答案。通过持续练习，这些技巧将变得得心应手，为你学习不等式、函数和微积分等更高阶内容打下坚实基础。坚持练习，祝你在 Edexcel 考试中取得优异成绩！
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