Mastering Quadratics for Edexcel IGCSE Maths | 精通Edexcel IGCSE数学二次函数

📚 Mastering Quadratics for Edexcel IGCSE Maths | 精通Edexcel IGCSE数学二次函数

Quadratics are one of the most essential topics in the Edexcel IGCSE Mathematics syllabus. In this article, we will explore every key skill you need, from factorising to graphing, with clear examples and exam-level tips.

二次函数是Edexcel IGCSE数学大纲中最核心的主题之一。本文将带你掌握所有关键技能，从因式分解到画图，配有清晰的例题和考试级技巧。

1. Quadratic Expressions | 二次表达式简介

A quadratic expression is an expression of the form ax² + bx + c, where x is a variable and a, b, c are constants with a ≠ 0. The highest power of x is 2, which gives the expression its characteristic ‘curved’ shape when plotted.

二次表达式是形如 ax² + bx + c 的代数式，其中 x 是变量，a、b、c 为常数且 a ≠ 0。x 的最高次幂为 2，这使得它绘成图像时呈现特有的“曲线”形状。

The coefficient a controls whether the curve opens upwards (a > 0) or downwards (a < 0).

系数 a 决定曲线开口朝上（a > 0）还是开口朝下（a < 0）。
The constant c represents the y-intercept of the curve.

常数 c 表示曲线与 y 轴的交点。

2. Expanding and Factorising Quadratics | 展开与因式分解二次式

Expanding involves multiplying two binomials, for example (x + 3)(x − 2) = x² + x − 6. This is often called the FOIL method. Factorising is the reverse process: rewriting x² + x − 6 back into (x + 3)(x − 2).

展开是指将两个二项式相乘，例如 (x + 3)(x − 2) = x² + x − 6，常用 FOIL 方法。因式分解是逆过程：将 x² + x − 6 还原为 (x + 3)(x − 2)。

(x + p)(x + q) = x² + (p + q)x + pq

When factorising x² + bx + c, look for two numbers that multiply to give c and add to give b.

因式分解 x² + bx + c 时，寻找两个数，它们相乘得 c，相加得 b。
For expressions like ax² + bx + c with a ≠ 1, use the ‘ac method’ or trial and error.

对于 a ≠ 1 的 ax² + bx + c，使用“ac 法”或尝试法。

3. Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程

A quadratic equation is typically written as ax² + bx + c = 0. If the left-hand side can be factorised, the solutions are obtained by setting each factor equal to zero.

二次方程通常写成 ax² + bx + c = 0。如果左边可以因式分解，则通过令每个因式为 0 来求得解。

If (px + m)(qx + n) = 0, then px + m = 0 or qx + n = 0

Example: Solve x² − 5x + 6 = 0. Factorise to (x − 2)(x − 3) = 0, so x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。

4. Solving Quadratic Equations by Completing the Square | 用配方法解二次方程

Completing the square rewrites a quadratic in the form a(x + p)² + q. This method works for all quadratics, even those that cannot be factorised.

配方法将二次式改写为 a(x + p)² + q 的形式。这种方法适用于所有二次式，包括不能因式分解的。

x² + bx = (x + b/2)² − (b/2)²

The vertex of the parabola is at (−p, q) when written as a(x + p)² + q.

当写成 a(x + p)² + q 时，抛物线的顶点坐标为 (−p, q)。
To solve an equation, rearrange to (x + p)² = k and then take the square root.

解方程时，整理成 (x + p)² = k，然后两边开平方。

5. The Quadratic Formula | 二次公式

The quadratic formula gives solutions to any quadratic equation ax² + bx + c = 0. It is derived from completing the square and is essential for non-factorisable cases.

二次公式给出了任意二次方程 ax² + bx + c = 0 的解。它由配方推导而来，对不可因式分解的情况至关重要。

x = (−b ± √(b² − 4ac)) / (2a)

Substitute the values of a, b, c carefully, especially noting signs.

代入 a、b、c 的值时要仔细，特别注意符号。
Wrap the numerator in brackets to avoid sign errors when using a calculator.

使用计算器时，把分子用括号括起来，避免符号错误。

6. The Discriminant | 判别式

The discriminant is the part b² − 4ac inside the quadratic formula. It tells us how many real roots a quadratic equation has without solving it fully.

判别式是二次公式中的 b² − 4ac 部分。它无需完全解方程，就能告诉我们二次方程实根的个数。

Δ = b² − 4ac

If Δ > 0, there are two distinct real roots.

若 Δ > 0，方程有两个不同的实根。
If Δ = 0, there is one repeated real root.

若 Δ = 0，方程有一个重实根。
If Δ < 0, there are no real roots.

若 Δ < 0，方程没有实根。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of a quadratic function is a parabola. Its shape, orientation and position are determined by the coefficients a, b and c.

二次函数的图像是抛物线。其形状、开口方向和位置由系数 a、b、c 决定。

The y-intercept is always c.

与 y 轴的交点始终是 c。
The roots of the equation correspond to the x-intercepts of the graph.

方程的根对应图像与 x 轴的交点。
The vertex can be found by completing the square or using x = −b/(2a).

顶点可通过配方或使用 x = −b/(2a) 求得。

Axis of symmetry: x = −b / (2a)

8. Solving Quadratic Inequalities | 解二次不等式

To solve a quadratic inequality like x² − 5x + 6 > 0, first find the roots, then test intervals on a number line to determine where the inequality holds.

解形如 x² − 5x + 6 > 0 的二次不等式时，先求根，然后在数轴上测试区间，确定满足不等式的范围。

For a > 0, the quadratic is positive outside the roots and negative between them.

当 a > 0 时，二次式在两根之外为正，两根之间为负。
Remember to include or exclude the roots depending on whether the inequality is strict (< or >) or inclusive (≤ or ≥).

根据不等式是严格（< 或 >）还是含等（≤ 或 ≥）来决定是否包含根。

9. Applications in Problem Solving | 在实际问题中的应用

Quadratics appear in many real-world contexts, such as projectile motion, area problems and profit optimisation. Word problems often lead to quadratic equations that need solving.

二次函数出现在许多实际情境中，例如抛体运动、面积问题和利润优化。应用题通常会导致需要解二次方程。

Example: A rectangle has length (x + 5) cm and width (x − 2) cm. Its area is 36 cm². Find x.

例：一个矩形长 (x + 5) cm，宽 (x − 2) cm，面积为 36 cm²。求 x。

(x + 5)(x − 2) = 36 → x² + 3x − 10 = 36 → x² + 3x − 46 = 0

Use the quadratic formula to get x ≈ 5.19 (ignoring the negative solution).

使用二次公式得 x ≈ 5.19（忽略负解）。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

In the Edexcel IGCSE exam, many students lose marks on simple arithmetic or sign errors. A few careful habits can make a big difference.

在Edexcel IGCSE考试中，许多学生因简单的算术或符号错误丢分。几个仔细的习惯会带来很大不同。

Always write the equation in the form ax² + bx + c = 0 before solving.

求解前始终将方程写成 ax² + bx + c = 0 的形式。
Check your factorisation by expanding it mentally.

通过心算展开来检查你的因式分解是否正确。
When using the quadratic formula, compute the discriminant first to see if solutions exist.

使用二次公式时，先计算判别式，判断解是否存在。
If a question says ‘give your answer to 2 decimal places’, do not leave surds.

如果题目要求“保留两位小数”，不要留下根号形式。
Sketch the graph of the quadratic to check the reasonableness of your answer.

画出二次函数草图，检查答案是否合理。

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Mastering Quadratics for Edexcel IGCSE Maths | 精通Edexcel IGCSE数学二次函数