Mastering Quadratic Equations and Graphs: A Guide for Edexcel IGCSE | 掌握二次方程与图像：Edexcel IGCSE 备考指南

📚 Mastering Quadratic Equations and Graphs: A Guide for Edexcel IGCSE | 掌握二次方程与图像：Edexcel IGCSE 备考指南

Quadratic equations are a cornerstone of the Edexcel IGCSE Mathematics syllabus. This revision guide breaks down the essential concepts, from standard form and solving techniques to graphing and the discriminant, ensuring you have a solid foundation for the exam.

二次方程是 Edexcel IGCSE 数学教学大纲的基石。本复习指南详细解析了核心概念，从标准形式、求解技巧到函数图像和判别式，确保你为考试打下坚实的基础。

1. The Standard Form | 标准形式

A quadratic equation is any polynomial equation where the highest power of the variable is 2. The general standard form is written as: ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

二次方程是变量的最高次幂为 2 的任意多项式方程。其一般标准形式写作：ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

• ‘a’ determines the width and direction of the parabola (curve).

  ‘a’ 决定抛物线的宽度和开口方向。

• ‘b’ affects the position of the axis of symmetry.

  ‘b’ 影响对称轴的位置。

• ‘c’ is the y-intercept (where the curve crosses the y-axis).

  ‘c’ 是 y 轴截距（曲线与 y 轴相交的点）。

2. Solving by Factorisation | 求解方法一：因式分解

Factorisation is often the quickest method if the quadratic expression factorises neatly. The principle lies in the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.

因式分解通常是解决能整齐分解的二次方程的最快方法。其原理在于零乘积性质：如果两个因子的乘积为零，则至少有一个因子必须为零。

For example, solve x² – 5x + 6 = 0. Factorising the left-hand side gives (x – 2)(x – 3) = 0. Therefore, x = 2 or x = 3.

例如，求解 x² – 5x + 6 = 0。将左侧因式分解得 (x – 2)(x – 3) = 0。因此，x = 2 或 x = 3。

• Step 1: Rearrange the equation into the standard form ax² + bx + c = 0.

  步骤 1：将方程整理为标准形式 ax² + bx + c = 0。

• Step 2: Factorise the left-hand side into two brackets.

  步骤 2：将左侧分解为两个括号的乘积。

• Step 3: Set each bracket equal to zero.

  步骤 3：令每个括号等于零。

• Step 4: Solve the resulting linear equations to find two roots.

  步骤 4：解所得的线性方程，找到两个根。

3. The Quadratic Formula | 二次公式

When factorisation is difficult or impossible, the quadratic formula provides a universal method. For any quadratic equation ax² + bx + c = 0, the solutions are given by the formula:

当因式分解困难或无法进行时，二次公式提供了一种通用解法。对于任意二次方程 ax² + bx + c = 0，其解由以下公式给出：

x = (-b ± √(b² – 4ac)) / (2a)

You must memorise this formula as it is not provided in the Edexcel IGCSE formula sheet. Always write it down clearly before substituting values to avoid arithmetic errors.

你必须牢记这个公式，因为它不会出现在 Edexcel IGCSE 的公式表中。务必先清晰写下公式，再代入数值，以避免算术错误。

4. Completing the Square | 配方法

Completing the square is another algebraic technique used to solve quadratics and to find the turning point of a parabola. The key identity for this method is:

配方法是另一种解决二次方程并找到抛物线转折点的代数技术。此方法的关键恒等式是：

x² + bx = (x + b/2)² – (b/2)²

For example, solve x² + 6x + 2 = 0 by completing the square. First, rewrite the first two terms: x² + 6x = (x + 3)² – 9. The full equation becomes (x + 3)² – 9 + 2 = 0, which simplifies to (x + 3)² = 7.

例如，用配方法求解 x² + 6x + 2 = 0。首先，重写前两项：x² + 6x = (x + 3)² – 9。整个方程变为 (x + 3)² – 9 + 2 = 0，化简得 (x + 3)² = 7。

Taking the square root of both sides gives x + 3 = ±√7. Thus, the exact solutions are x = -3 ± √7.

对方程两边开平方得 x + 3 = ±√7。因此，精确解为 x = -3 ± √7。

5. The Discriminant | 判别式（Δ）

The discriminant, denoted by the Greek letter Δ, is the part of the quadratic formula under the square root sign: Δ = b² – 4ac. It tells us the nature of the roots without fully solving the equation.

判别式，用希腊字母 Δ 表示，是二次公式中根号下的部分：Δ = b² – 4ac。它能在不解完全方程的情况下告诉我们根的性质。