Example 2.6.2 Solving a Quadratic Equation Using the Quadratic Formula | 例 2.6.2：使用求根公式解二次方程

📚 Example 2.6.2 Solving a Quadratic Equation Using the Quadratic Formula | 例 2.6.2：使用求根公式解二次方程

In this AQA A-Level Mathematics worked example, we solve the quadratic equation x² − 4x + 1 = 0 by using the quadratic formula. The solution is left in exact surd form, which is the standard requirement for non-calculator questions.

在这个 AQA A-Level 数学例题中，我们使用求根公式解二次方程 x² − 4x + 1 = 0。答案用最简根式表示，这是非计算器题目的标准要求。

1. The Worked Example | 例题

The example asks us to solve x² − 4x + 1 = 0, giving the answers in exact surd form. This means we cannot simply use a calculator to obtain rounded decimal answers.

本例题要求解方程 x² − 4x + 1 = 0，并用精确根式给出答案。这意味着我们不能只用计算器得到四舍五入的小数答案。

This type of question appears regularly in AQA AS and A-Level papers, especially in the non-calculator section where exact methods are tested.

这类题目在 AQA AS 和 A-Level 试卷中经常出现，尤其是在非计算器部分，考查精确方法。

2. Why We Use the Quadratic Formula | 为什么使用求根公式

Factorising x² − 4x + 1 = 0 is not obvious because there are no integer factors of 1 that add up to −4. The only factor pair of 1 is 1 and 1, and their sum is 2, not −4.

x² − 4x + 1 = 0 不容易因式分解，因为没有整数因子 1 能相加得到 −4。1 的唯一因子对是 1 和 1，它们的和是 2，不是 −4。

The quadratic formula always works for any quadratic equation, so it is the safest method here. Completing the square would also work, and we will check it later as an alternative.

求根公式对任意二次方程都适用，因此这里是更稳妥的方法。配方法同样可行，我们稍后会作为替代方法进行检验。

3. The Quadratic Formula | 求根公式

For any quadratic equation of the form ax² + bx + c = 0, the roots are given by:

对于任意形如 ax² + bx + c = 0 的二次方程，其根由以下公式给出：

x = (−b ± √(b² − 4ac)) / 2a

The symbol ± means that there are usually two solutions: one with a plus sign and one with a minus sign. This formula is provided in the A

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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