📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations form the backbone of many IGCSE Mathematics topics. Whether you are aiming for a grade 4 or pushing for a 9, the ability to solve quadratics fluently will unlock marks across algebra, graphs, and problem‑solving questions. This article covers every essential method – factorising, completing the square, and the quadratic formula – while linking each to the Edexcel specification. Read on to turn a potentially tricky topic into a secure strength.
二次方程是 IGCSE 数学许多主题的基石。不论你的目标是 4 分还是冲刺 9 分，熟练求解二次方程都能帮你在代数、图像和应用题中稳稳拿分。本文涵盖所有核心方法——因式分解、配方法和求根公式，并紧密贴合 Edexcel 考纲。仔细阅读，把这一个可能棘手的主题变成你的强项。
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 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. In IGCSE Edexcel, you will meet quadratics in purely algebraic settings as well as in contexts such as area problems, projectile motion and profit modelling.
二次方程是任意能写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数为 2，因此其图像呈特有的 U 形，称为抛物线。在 Edexcel IGCSE 考试中，你不仅会在纯代数题中遇到二次方程，还会在面积问题、抛体运动和利润建模等情境中见到它。
2. Standard Form and Coefficients | 标准形式与系数
Before solving, always rearrange the equation into ax² + bx + c = 0. The coefficient a cannot be zero, but b or c may be zero, giving special cases like x² – correctly is crucial for later methods such as the quadratic formula. For example, in 3x² – 5x + 2 = 0, a = 3, b = –5 and c = 2; careful handling of signs prevents errors.
求解之前，务必先将方程整理成 ax² + bx + c = 0 的形式。系数 a 不能为零，但 b 或 c 可以为零，从而得到如 x² – 9 = 0 的特殊情形。正确识别 a、b 和 c 对后面使用求根公式等方法至关重要。例如在 3x² – 5x + 2 = 0 中，a = 3，b = –5，c = 2；谨慎处理符号可以避免错误。
3. Solving by Factorising (Part 1) | 通过因式分解求解（第一部分）
two numbers that multiply to 6 and add to 5: 2 and 3. Thus (x + 2)(x + 3) = 0, giving x = –2 or x = – when a = 1 and the roots are rational. Always set each bracket to zero after factorising.
当一个二次方程能写成两个一次括号的乘积时，求解就很简单。对于 x² + 5x + 6 = 0，找出两个乘积为 6、和为 5 的数：2 和 3。于是得到 (x + 2)(x + 3) = 0，从而 x = –2 或 x = –3。当 a = 1 且根为有理数时，该方法非常高效。因式分解后，务必令每个括号分别等于零。
4. Factorising When a > 1 | 当 a > 1 时的因式分解
+ 3 = 0. Factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. Solutions are x = –½ and x = –3. Practice this technique until it becomes automatic.
对于 2x² + 7x + 3 = 0 这样的二次三项式，分解步骤稍多。先将 a 与 c 相乘：2 × 3 = 6。找出乘积为 6、和为 7 的两个数：6 和 1。将中间项拆开：2x² + 6x + x + 3 = 0。分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。解为 x = –½ 和 x = –3。反复练习，直到该技巧成为本能。
5. Solving by Completing the Square | 配方法求解
² + bx into a perfect square trinomial. Take x² + 6x + 5 = 0. Move the constant: x² + 6x = – h sides: x² + 6x + 9 = 4, so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, hence x = –1 or x = – his method is particularly useful for deriving the quadratic formula and sketching graphs.
配方法把 x² + bx 转化为完全平方式。以 x² + 6x + 5 = 0 为例。移常数项：x² + 6x = –5。两边同时加上 (b/2)² = 9，得 x² + 6x + 9 = 4，即 (x + 3)² = 4。开平方得 x + 3 = ±2，因此 x = –1 或 x = –5。该方法在推导求根公式和绘制图像时尤为有用。
6. Deriving the Quadratic Formula | 推导二次公式
bx + c = 0, divide through by a: x² + (b/a)x + c/a = 0. Complete the square: x² + (b/a)x = –c/a. Add (b/2a)²: x² + (b/a)x + b²/4a² = b²/4a² – c/a. The left side becomes (x + b/2a)², and the right side simplifies to (b² – 4ac)/4a². Taking square roots and isolating x yields:
x = [ –b ± √b² – 4ac ] / (2a)
从 ax² + bx + c = 0 出发，两边除以 a：x² + (b/a)x + c/a = 0。配方：x² + (b/a)x = –c/a。加上 (b/2a)²：x² + (b/a)x + b²/4a² = b²/4a² – c/a。左边变成 (x + b/2a)²，右边化简为 (b² – 4ac)/4a²。开方并移项即得：
x = [ –b ± √b² – 4ac ] / (2a)
7. Using the Quadratic Formula | 使用二次公式
formula solves any quadratic equation, even when factorisation is impossible. For 3x² + 4x – 2 = 0, identify a = 3, b = 4, c = –2. Substitute into the formula: x = [ –4 ± √(4² – 4×3×(–2)) ] / (2×3) = [ –4 ± √16 + 24 ] / 6 = [ –4 ± √40 ] / 6. Simplify √40 = 2√10, so x = [ –4 ± 2√10 ] / 6 = [ –2 ± √10 ] / 3. Always simplify surds for full marks.
求根公式能解任何二次方程，即使无法因式分解也能求解。对于 3x² + 4x – 2 = 0，确定 a = 3, b = 4, c = –2。代入公式：x = [ –4 ± √(4² – 4×3×(–2)) ] / (2×3) = [ –4 ± √16 + 24 ] / 6 = [ –4 ± √40 ] / 6。化简 √40 = 2√10，所以 x = [ –4 ± 2√10 ] / 6 = [ –2 ± √10 ] / 3。务必化简根式以拿到满分。
8. Discriminant and Nature of Roots | 判别式与根的性质
The expression b² – 4ac inside the square root is called the discriminant Δ. It determines how many real solutions exist:
- Δ > 0: two distinct real roots.
- Δ = 0: exactly one real root (a repeated root).
- Δ < 0: no real roots (the parabola does not cross the x‑axis).
根号内的表达式 b² – 4ac 叫做判别式 Δ。它决定了方程有多少个实数解：
- Δ > 0：两个不同的实数根。
- Δ = 0：恰好一个实数根（重根）。
- Δ < 0：无实数根（抛物线不与 x 轴相交）。
9. Graphical Interpretation | 图形解释
uation y = ax² + bx + c represents a parabola. The solutions to ax² + bx + c = 0 are the x‑ f the graph. Completing the square reveals the vertex: y = a(x + p)² + q has a vertex at (–p, q). When a > 0 the parabola opens upwards (∪), when a < 0 it opens downwards (∩ ween algebra and graph helps with sketch questions and identifying maximum/minimum values.
二次方程 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是图像与 x 轴的交点。通过配方可以得到顶点坐标：y = a(x + p)² + q 的顶点为 (–p, q)。当 a > 0 时抛物线开口向上（∪），当 a < 0 时开口向下（∩）。理解代数与图像的关联有助于解答草图题和识别最大值或最小值。
10. Word Problems and Applications | 应用题与实际应用
IGCSE frequently embeds quadratics in real‑ 0, solve using the most appropriate method, and finally interpret the solution in context – rejecting negative lengths or times that don’t make sense.
IGCSE 常把二次方程融入实际情境。典型例子包括：已知面积求矩形边长，计算抛出的小球何时落地（令高度 = 0），或优化二次函数代表的利润。务必清晰定义变量，列出 ax² + bx + c = 0 形式的方程，选用最合适的方法求解，最后结合实际解释答案——舍去不合理的负长度或负时间。
11. Common Mistakes to Avoid | 常见错误避免
Watch out for these frequent pitfalls:
- Forgetting to set the equation to zero before factorising.
- Misidentifying a, b, c with wrong signs, especially when b is negative.
- Dividing by x and losing a root – always bring all terms to one side.
- Incorrectly simplifying surds or fractions in the quadratic formula.
- Forgetting the ± when taking square roots.
- Not checking whether solutions fit the context of the word problem.
请当心以下常见失分点：
- 因式分解前忘记先将方程化为“等于零”的形式。
- 识别 a、b、c 时搞错符号，尤其是 b 为负数时。
- 两边同除以 x 导致丢根——应将所有项移到同一侧。
- 在求根公式中化简根式或分数时出错。
- 开平方时漏写 ± 号。
- 没有检查解是否符合应用题的实际背景。
12. Exam Tips and Summary | 考试技巧与总结
When facing a quadratic in an Edexcel IGCSE exam: (1) Read the question carefully to see whether a specific method is demanded. (2) If factorising looks difficult, skip directly to the formula – it always works. (3) Always simplify your final answers. (4) Use the discriminant to check the reasonableness of your solutions. (5) In graph or word problems, present your answer in the context of the question, with correct units. Build confidence by practising mixed exercises until selecting the fastest method becomes second nature. A solid command of quadratics will not only secure high marks but also provide a smooth transition to A‑level Mathematics.
在 Edexcel IGCSE 考试中面对二次方程时：(1) 仔细读题，看是否要求特定解法。(2) 若因式分解不太顺手，直接使用求根公式——它总是有效的。(3) 始终化简最终答案。(4) 用判别式检查解的合理性。(5) 在图像题或应用题中，结合题目情境给出答案，并带上正确单位。通过混合练习建立信心，直到选择最快解法成为本能。扎实掌握二次方程不仅能在考试中拿下高分，还能为 A‑level 数学的平稳过渡做好铺垫。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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