Solving Quadratic Equations and Functions | 二次方程与函数的完全指南

📚 Solving Quadratic Equations and Functions | 二次方程与函数的完全指南

This comprehensive revision guide covers every core aspect of quadratic equations and functions required for IGCSE Mathematics. You will learn the standard form, solution methods, the discriminant, graphs, and real-world applications, with clear step-by-step explanations.

本复习指南涵盖 IGCSE 数学中二次方程与函数的全部核心考点。你将学习标准形式、求解方法、判别式、图像以及实际应用，并获得清晰的分步讲解。

1. The Standard Form | 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 and a ≠ 0.

二次方程可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。

In the equation 3x² – 2x + 5 = 0, the coefficient a is 3, b is -2, and c is 5.

在方程 3x² – 2x + 5 = 0 中，系数 a 为 3，b 为 -2，c 为 5。

A quadratic expression has degree 2 because the highest power of x is x². If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

二次表达式的次数为 2，因为 x 的最高次幂是 x²。若 a = 0，方程就变成了一次方程，因此 a ≠ 0 这一条件至关重要。

2. Solving by Factorisation | 因式分解法

To solve a quadratic by factorisation, first write the equation in the form ax² + bx + c = 0. Then factorise the left-hand side into two linear factors.

用因式分解法解二次方程时，首先将方程写成 ax² + bx + c = 0 的形式，然后把左边分解成两个一次因式的乘积。

For example, for x² – 5x + 6 = 0, we look for two numbers whose product is 6 and whose sum is -5. These numbers are -2 and -3, so (x – 2)(x – 3) = 0.

例如，对于 x² – 5x + 6 = 0，我们需要找到两个数，它们的乘积为 6，和为 -5。这两个数是 -2 和 -3，因此 (x – 2)(x – 3) = 0。

Using the zero product property, if the product of two factors is zero, at least one factor must be zero. Hence x = 2 or x = 3.

根据零乘积性质，若两个因式的乘积为零，则至少有一个因式为零。因此 x = 2 或 x = 3。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is useful when factorisation is difficult.

配方法可将二次式改写为 (x + p)² + q 的形式。当因式分解较困难时，这一方法非常有用。

Start with x² + bx. Add and subtract (b/2)² to form a perfect square: x² + bx + (b/2)² = (x + b/2)².

从 x² + bx 开始，加减 (b/2)² 来构造完全平方：x² + bx + (b/2)² = (x + b/2)²。

For example, x² + 6x + 2 = 0 becomes (x + 3)² – 7 = 0 because 6x has b = 6, so (b/2)² = 9.

例如，x² + 6x + 2 = 0 可化为 (x + 3)² – 7 = 0，因为 6x 中 b = 6，所以 (b/2)² = 9。

Then solve by isolating the square: (x + 3)² = 7, so x + 3 = ±√7, giving x = -3 ± √7.

然后通过分离平方项求解：(x + 3)² = 7，所以 x + 3 = ±√7，得到 x = -3 ± √7。

4. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:

对于任何

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