📚 253. Solving Quadratic Equations | 253. 解二次方程
Quadratic equations lie at the heart of the Edexcel IGCSE Mathematics syllabus, appearing in algebraic manipulation, graph sketching, and real‑world problem‑solving. Whether you are sitting the Foundation or Higher tier, a solid understanding of how to solve quadratics by factorising, completing the square, applying the quadratic formula, and interpreting the discriminant is essential for top marks. This article breaks down every key method, highlights common pitfalls, and provides exam‑focused tips to help you master the topic with confidence.
二次方程是 Edexcel IGCSE 数学考纲的核心内容，广泛出现在代数运算、函数图像绘制和实际应用问题中。无论参加基础级还是进阶级考试，扎实掌握因式分解法、配方法、二次公式法以及判别式的意义，都是取得高分的关键。本文将逐一拆解每种求解方法，指出常见错误，并提供紧扣考试的技巧，帮助你自信地攻克这一主题。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable (usually x) is squared. In its general form, it is written as ax² + bx + c = 0 tion becomes linear. Quadratics can have two solutions, one solution, or no real solutions – a property governed by the discriminant.
二次方程是指最高次数为 2 的多项式方程，即未知数（通常为 x）的最高次幂是平方项。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，方程就退化为一次方程。二次方程可能有两个解、一个解或无实数解——这一性质由判别式决定。
2. Standard Form and Coefficient Identification | 标准形式与系数识别
0. Identify the coefficients a, b and c carefully, including their signs. For example, in 2x² − 5x + 3 = 0, a = 2, b = −5 and c = 3. In 4x − x² = 6 c = −6. Mistakes often occur when the sign of b or c is ignored, so highlight them early.
在使用任何解法之前，务必先将方程整理为标准形式 ax² + bx + c = 0。仔细识别系数 a、b、c，并注意它们的正负号。例如在 2x² − 5x + 3 = 0 中，a = 2，b = −5，c = 3。对于 4x − x² = 6，需先改写成 −x² + 4x − 6 = 0，得到 a = −1，b = 4，c = −6。很多错误源于忽略 b 或 c 的符号，因此要格外留意。
3. Solving by Factorising (Monic Quadratics) | 因式分解法求解（首一二次式）
orises to (x + 2)(x + 3) = 0, so x = −2 or x = −3. Always check by expanding the brackets.
当 a = 1 时，二次式称为首一的：x² + bx + c = 0。因式分解需要寻找两个数 p、q，使得 p + q = b 且 pq = c。方程便可写成 (x + p)(x + q) = 0，从而得到解 x = −p 或 x = −q。例如，x² + 5x + 6 = 0 可分解为 (x + 2)(x + 3) = 0，故 x = −2 或 x = −3。务必通过展开括号来检验。
4. Solving by Factorising (Non‑Monic Quadratics) | 因式分解法求解（非首一二次式）
For ax² + bx + c with a ≠ 1, you can use the ‘ Rewrite as 2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3). Solutions: x = −½ or x = −3.
对于 a ≠ 1 的 ax² + bx + c，可使用“ac 法”或有系统的试凑。先计算 a 与 c 的积 ac，再找出两个数，它们的积为 ac，和为 b。按此拆分中间项，然后分组分解。例如 2x² + 7x + 3，ac = 6，可找到 6 和 1，因为 6+1=7。重写为 2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)。解为 x = −½ 或 x = −3。
5. Completing the Square | 配方法
2x² + 8x − 5 = 0 → 2(x² + 4x) − 5 = 0 → 2[(x+2)² − 4] − 5 = 0 → 2(x+2)² − 13 = 0 → x+2 = ±√13/2. This is also how the quadratic formula is derived.
配方法将二次式改写为 a(x + p)² + q = 0 的形式，从而直接求出 x。对于 x² + bx，完全平方式为 (x + b/2)²，展开得 x² + bx + (b/2)²。因此 x² + bx = (x + b/2)² − (b/2)²。若 a ≠ 1，先提取 a。例如 2x² + 8x − 5 = 0 → 2(x² + 4x) − 5 = 0 → 2[(x+2)² − 4] − 5 = 0 → 2(x+2)² − 13 = 0 → x+2 = ±√13/2。二次公式正是由此推导而出。
6. The Quadratic Formula | 二次公式法
For any quadratic ax² + bx + c = 0, the solutions are given by:
x = (−b ± √b² − 4ac) / (2a)
obtain two solutions. Always simplify the square root where possible and reduce fractions.
对于任意 ax² + bx + c = 0，其解由以下公式给出：
x = (−b ± √b² − 4ac) / (2a)
Edexcel IGCSE 进阶级考试会提供该公式，但你必须会正确使用。代入 a、b、c 的值（注意符号），计算判别式 b² − 4ac，再通过 ± 得到两个解。尽量化简根号并约分。
7. The Discriminant and Nature of Roots | 判别式与根的性质
discriminant, reveals how many real solutions exist without solving the equation. The table below summarises its significance:
| Discriminant (D = b² − 4ac) | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | One repeated real root (tangent to x‑axis) |
| D < 0 | No real roots (graph does not cross x‑axis) |
l solutions. This tool is invaluable for checking answers and solving problems where the exact roots are not required.
表达式 D = b² − 4ac 称为判别式，无需解方程就能判断实数解的情况。下表概括其意义：
| 判别式 (D = b² − 4ac) | 根的性质 |
|---|---|
| D > 0 | 两个相异实根 |
| D = 0 | 一个重复实根（与 x 轴相切） |
| D < 0 | 无实根（图像不与 x 轴相交） |
例如 x² − 6x + 9 = 0 的 D = 0 → 唯一根 x = 3。2x² + 4x + 5 = 0 的 D = −24 → 无实数解。判别式在验证答案和处理无需精确求根的问题时十分有用。
8. Graphical Interpretation of Roots | 根的图像解释
= ax² + bx + c produces a parabola. The solutions to ax² + bx + c = 0 correspond to the x‑intercepts of the graph. If D > 0, the curve cuts the x‑axis at two points. If D = 0, the vertex lies on the x‑axis. If D < 0, the entire curve stays above or below the x‑ why some equations have no real solutions and reinforces the meaning of the discriminant.
二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解即为图像与 x 轴交点的横坐标。若 D > 0，曲线与 x 轴相交于两点；若 D = 0，顶点恰好落在 x 轴上；若 D < 0，整条曲线完全位于 x 轴的上方或下方，具体取决于 a 的正负。画图有助于直观理解为什么某些方程无实数解，并加深对判别式的理解。
9. Solving Quadratic Inequalities | 解二次不等式
solved by sketching the parabola and identifying intervals where the curve is below the x‑ sent answer with strict attention to < or ≤ and consider the direction of the parabola.
形如 x² − 4x + 3 < 0 的二次不等式可通过绘制抛物线并找出曲线在 x 轴下方的区间来求解。首先求出根（x = 1 和 x = 3），注意到图像开口向上（a > 0），因此该式在两根之间为负值，解为 1 < x < 3。对于 ax² + bx + c > 0，解在两根之外。务必严格区分 < 或 ≤，并考虑抛物线开口方向。
10. Common Mistakes to Avoid | 常见错误
elling terms incorrectly (never divide by a variable); and not checking whether factors actually expand to the original expression. In inequalities, remember to flip the sign only when multiplying or dividing by a negative number, not when simply solving quadratic factors. Always re‑substitute your solutions to verify them.
许多失分源于可避免的错误：在分解前忘记将方程设为零；错误识别 b 和 c 的符号；使用公式时遗漏 ± 号；错误地约去含 x 的项（绝不能除以变量）；没有验证因式展开后是否与原式一致。在不等式中，只有在乘以或除以负数时才需要不等号变向，而不是在解二次因式时随意变号。务必代入求解结果进行检验。
11. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧
When tackling quadratic questions in Edexcel IGCSE papers, first check if the equation can be factorised quickly – this is often the quickest method for integer solutions. If factorisation is awkward, fall back on the quadratic formula or completing the square, especially when coefficients are large or non‑integer. Show each step clearly; even if the final answer is wrong, method marks are available. Keep an eye on the discriminant to confirm the number of solutions expected. If the question asks for answers to a specific number of decimal places or in surd form, present them exactly as requested.
在解答 Edexcel IGCSE 试卷中的二次方程题目时，先检查能否快速因式分解——对整数解通常这是最快的方法。若不易分解，再使用二次公式或配方法，特别是当系数较大或为非整数时。清晰展示每一步骤，即使最终答案出错，也有方法分。留意判别式以确认预期解的个数。如果题目要求将答案保留到指定小数位或以根式呈现，必须严格按照要求作答。
12. Summary and Key Formulae | 总结与关键公式
he nature of roots. Regular practice with past papers builds speed and confidence, ensuring you can select the most efficient method for any question. Remember the formula:
x = (−b ± √b² − 4ac) / (2a)
掌握二次方程需要熟练运用四种主要策略：因式分解、配方法、二次公式和图像分析。牢记标准形式 ax² + bx + c = 0，准确识别系数，理解 D = b² − 4ac 如何决定根的性质。通过历年真题反复练习可以提升速度和自信，确保在任何题目中都能选用最高效的方法。记住公式：
x = (−b ± √b² − 4ac) / (2a)
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