Mixed Exercise 2: Quadratics and Algebraic Techniques | 混合练习2：二次函数与代数技巧

📚 Mixed Exercise 2: Quadratics and Algebraic Techniques | 混合练习2：二次函数与代数技巧

Mixed Exercise 2 in the Edexcel A Level Mathematics course typically brings together the core skills from the quadratics chapter: solving quadratic equations, using the discriminant, completing the square, handling inequalities, and applying these ideas to curves and real-world problems. This revision article walks through the key methods, common pitfalls, and exam-style strategies you need to master the exercise with confidence.

Edexcel A Level 数学课程中的 Mixed Exercise 2 通常整合了二次函数一章的核心技能：解二次方程、使用判别式、配方法、处理不等式，以及将这些思想应用于曲线和实际问题。本文带你梳理关键方法、常见错误和考试风格策略，帮助你有把握地完成练习。

1. Solving Quadratic Equations | 解二次方程

In Edexcel A Level Mathematics, Mixed Exercise 2 often opens with straightforward quadratic equations. You should be able to solve them by factorising, by completing the square, or by applying the quadratic formula.

在 Edexcel A Level 数学中，Mixed Exercise 2 通常从基础的二次方程开始。你应当能够通过因式分解、配方法或求根公式来解这些方程。

The general quadratic equation is ax² + bx + c = 0, where a ≠ 0. If the expression factorises over integers, factorisation is the fastest route, but the quadratic formula always works.

一般二次方程为 ax² + bx + c = 0，其中 a ≠ 0。如果表达式可以在整数范围内因式分解，因式分解是最快的方法，但求根公式始终有效。

x = (-b ± √(b² – 4ac)) / 2a

For example, for x² – 7x + 10 = 0, factorising gives (x – 2)(x – 5) = 0, so x = 2 or x = 5.

例如，对于 x² – 7x + 10 = 0，因式分解得到 (x – 2)(x – 5) = 0，因此 x = 2 或 x = 5。

When the quadratic does not factorise, use the formula or complete the square. Remember to simplify surds and to state answers exactly unless a decimal approximation is requested.

当二次式不能因式分解时，使用公式或配方法。记住要化简根式，除非题目要求近似值，否则应给出精确答案。

2. Completing the Square | 配方法

Completing the square is essential for deriving the quadratic formula and for finding the vertex of a parabola. For a monic quadratic x² + bx, the completed square form is:

配方法对于推导求根公式和求抛物线的顶点至关重要。对于首项系数为 1 的二次式 x² + bx，配方法的完成形式为：

x² + bx = (x + b/2)² – (b/2)²

For a general quadratic ax² + bx + c, first factor a from the x terms, then complete the square inside the bracket.

对于一般二次式 ax² + bx + c，首先从 x 项中提取系数 a，然后在括号内完成配方。

This form reveals the vertex of y = ax² + bx + c as (-b/(2a), c – b²/(4a)). It also helps solve equations when factorising is not possible.

这种形式揭示了 y = ax² + bx + c 的顶点为 (-b/(2a), c – b²/(4a))。它也有助于在无法因式分解时解方程。

Example: x² + 6x + 1 = 0 becomes (x + 3)² – 8 = 0, so x = -3 ± 2√2.

示例：x² + 6x + 1 = 0 可化为 (x + 3)² – 8 = 0，因此 x = -3 ± 2√2。

3. The Discriminant | 判别式

The discriminant Δ (or D) is defined as Δ = b² – 4ac for ax² + bx + c = 0. It determines the nature of the roots without solving the equation.

判别式 Δ（或 D）定义为 Δ = b² – 4ac，其中 ax² + bx + c = 0。它无需解方程即可判断根的性质。