Mastering Quadratic Equations for Edexcel IGCSE | 掌握Edexcel IGCSE二次方程

📚 Mastering Quadratic Equations for Edexcel IGCSE | 掌握Edexcel IGCSE二次方程

Quadratic equations appear in nearly every Edexcel IGCSE Maths paper. Whether you are solving for an unknown, sketching a graph, or tackling a word problem, a solid grasp of the techniques is essential for securing top marks. This revision guide breaks down every method you need, explains when to use each one, and highlights common traps that cost students valuable marks.

二次方程几乎出现在Edexcel IGCSE数学的每一份试卷中。无论是求未知数、画函数图像还是解答应用题，扎实掌握相关方法都是获得高分的关键。本复习指南将详细拆解所有必需的方法，说明每种方法的适用场景，并指出常见失分陷阱。

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation can be written in the standard form ax² + bx + c = 0, where a, b and c are constants, with a ≠ 0. The degree of the equation is 2, which means the highest power of x is 2. If a = 0, the equation becomes linear, not quadratic.

二次方程可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。该方程的次数为 2，即 x 的最高次幂为 2。若 a = 0，则方程退化为一次方程，而非二次方程。

Before solving, always rearrange the equation so that all non-zero terms are on one side, leaving zero on the other. This is essential for both factorisation and the quadratic formula.

在求解前，务必整理方程，使所有非零项移到一边，另一边为零。这对于因式分解法和二次公式法都是至关重要的前提。

2. Solving by Factorisation | 因式分解法求解

Factorisation is often the fastest method when the quadratic has simple integer roots. To solve ax² + bx + c = 0, look for two numbers that multiply to give ac and add to give b, then split the middle term and factor by grouping.

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。求解 ax² + bx + c = 0 时，需找到两个数，使其乘积等于 ac、和等于 b，然后拆中项并分组分解。

For example, solve x² + 5x + 6 = 0. Find two numbers that multiply to 6 and add to 5: they are 2 and 3. So (x + 2)(x + 3) = 0. Therefore x = -2 or x = -3.

例如，解 x² + 5x + 6 = 0。找两个数，它们相乘得 6、相加得 5：即 2 和 3。于是 (x + 2)(x + 3) = 0，所以 x = -2 或 x = -3。

If the coefficient of x² is not 1, you may need to factor out a common factor first or use the ac method. Always check your factors by expanding them back.

如果 x² 的系数不为 1，你可能需要先提取公因数，或使用 ac 法（十字相乘法）。分解后务必展开验证。

3. Solving by Completing the Square | 配方法求解

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful when dealing with equations that cannot be easily factorised, or when you need to find the turning point of a parabola.

配方法将二次式改写为 a(x + p)² + q 的形式。它特别适用于难以因式分解的方程，也常用于求抛物线的顶点坐标。

For x² + bx + c, add and subtract (b/2)². For example, x² + 6x + 2 = 0 becomes (x + 3)² – 9 + 2 = 0, which simplifies to (x + 3)² = 7. Then x + 3 = ±√7, so x = -3 ± √7.

对于 x² + bx + c，加上并减去 (b/2)²。例如，x² + 6x + 2 = 0 可化为 (x + 3)² – 9 + 2 = 0，即 (x + 3)² = 7。于是 x + 3 = ±√7，所以 x = -3 ± √7。

When a ≠ 1, factor out a from the first two terms first, then complete the square inside the brackets.

当 a ≠ 1 时，先从前两项中提取 a，再对括号内部配方。

4. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation, including those with irrational or complex roots. Edexcel IGCSE provides this formula in the exam formula sheet, but you must know how to substitute and simplify correctly.

二次公式适用于一切二次方程，包括根为无理数或复数的情况。Edexcel IGCSE 的考纲公式表会给出此公式，但你必须掌握正确的代入与化简技巧。

x = (-b ± √(b² – 4ac)) / 2a

To use it, write the equation in standard form, identify a, b, and c, then substitute into the formula. Simplify the discriminant and the square root carefully.

使用步骤：先将方程写成标准形式，确定 a、b、c，然后代入公式，仔细化简判别式和根号。

For example, solve 2x² + 3x – 5 = 0. Here a = 2, b = 3, c = -5. Substitute to get x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4. Hence x = 1 or x = -5/2.

例如，解 2x² + 3x – 5 = 0。这里 a = 2，b = 3，c = -5。代入得 x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4，因此 x = 1 或 x = -5/2。

5. The Discriminant | 判别式

The discriminant, denoted by Δ = b² – 4ac, is part of the quadratic formula. It tells us how many real roots a quadratic equation has, without fully solving it.

判别式记作 Δ = b² – 4ac，是二次公式中的一部分。它用于判断二次方程实根的个数，而无需完整求解。

If Δ > 0, the equation has two distinct real roots.

若 Δ > 0，方程有两个不等的实数根。
If Δ = 0, the equation has exactly one real root (a repeated root).

若 Δ = 0，方程有一个实数根（重根）。
If Δ < 0, the equation has no real roots (only complex roots).

若 Δ < 0，方程没有实数根（仅有复数根）。

This is a quick check for sketching graphs and for inequality problems.

这在画图以及解不等式问题时是一种快速判断方法。

6. Graphs and Roots | 图像与根

The graph of y = ax² + bx + c is a parabola. The x-intercepts of the graph are exactly the real roots of the equation ax² + bx + c = 0.

函数 y = ax² + bx + c 的图像是抛物线。抛物线与 x 轴的交点正是方程 ax² + bx + c = 0 的实数根。

If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the parabola touches the axis at its vertex. If it is negative, the parabola does not touch the x-axis at all.

若判别式为正，抛物线穿过 x 轴两次；若为零，抛物线在顶点处与 x 轴相切；若为负，抛物线完全不接触 x 轴。

The axis of symmetry is given by x = -b/(2a), and the vertex has x-coordinate equal to this value. Completing the square reveals the vertex explicitly as (-p, q).

对称轴为 x = -b/(2a)，顶点的 x 坐标即为此值。通过配方可以直接得到顶点坐标 (-p, q)。

7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the relationships α + β = -b/a and αβ = c/a hold. These are derived from expanding a(x – α)(x – β).

对于二次方程 ax² + bx + c = 0，设其两根为 α 和 β，则满足 α + β = -b/a，αβ = c/a。这些关系由展开 a(x – α)(x – β) 推导得出。

These formulas are useful when you are given the roots and asked to form the quadratic equation, or when checking your solved values.

这些公式可用于已知根求作二次方程，也可用于验算已求得的根。

For example, if roots are 3 and -2, then sum = 1 and product = -6. The equation is x² – x – 6 = 0.

例如，若根为 3 和 -2，则和为 1，积为 -6，方程为 x² – x – 6 = 0。

8. Word Problems Involving Quadratics | 二次方程应用题

Quadratic word problems often involve areas, consecutive numbers, projectile motion, or profit maximisation. The key is to translate the English description into a quadratic equation.

二次方程应用题常涉及面积、连续整数、抛体运动或利润最大化等问题。关键在于把文字描述转化为二次方程。

For a rectangle problem: “The length of a rectangle is 3 cm greater than its width, and the area is 40 cm².” Let width = w, then length = w + 3. So w(w + 3) = 40, giving w² + 3w – 40 = 0. Factorise to (w + 8)(w – 5) = 0, so w = 5 (discard w = -8).

对于矩形问题：“一个矩形的长比宽多 3 cm，面积为 40 cm²。”设宽为 w，则长为 w + 3。于是 w(w + 3) = 40，即 w² + 3w – 40 = 0。因式分解得 (w + 8)(w – 5) = 0，因此 w = 5（舍去 w = -8）。

Always consider whether a negative solution is meaningful in the context. In many real-world problems, negative lengths or times are rejected.

务必判断负数解在语境中是否有意义。在许多实际问题中，负的长度或时间要被舍去。

9. Common Pitfalls and Exam Tips | 常见错误与考试技巧

Forgetting to rearrange the equation to zero before factorising or using the formula.

在因式分解或使用公式前忘记将方程整理为零。
Sign errors when substituting b into the quadratic formula, especially when b is negative.

将 b 代入二次公式时出现符号错误，尤其当 b 为负数时。
Losing a solution when taking square roots: remember both ± signs.

开平方时漏掉一个解：务必记住 ± 号。
Assuming every quadratic can be factorised; use the formula or completing the square when needed.

假设所有二次方程都能因式分解；必要时使用公式法或配方法。
Not checking whether the final answers satisfy the original equation.

不检查最终答案是否满足原方程。

In the exam, show all your working clearly. If a question asks for exact answers, leave square roots unsimplified unless instructed. If it asks for answers to a given degree of accuracy, use the calculator only as a final step.

考试中要清晰展示所有步骤。若题目要求精确值，除非另有说明，否则保留根号形式；若要求保留指定精度，仅在最后一步使用计算器。

Practise all four methods — factorisation, completing the square, formula, and graphical interpretation — until they feel automatic. Speed and accuracy in this topic will free up time for the rest of the paper.

练习这四种方法——因式分解法、配方法、公式法和图像解法——直到运用自如。在这个主题上的速度与准确性，将为你完成试卷其余部分节省宝贵时间。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Mastering Quadratic Equations for Edexcel IGCSE | 掌握Edexcel IGCSE二次方程