📚 Equations and Identities | 方程与恒等式
In A-Level Mathematics, mastering equations and identities is fundamental to problem-solving across algebra, trigonometry, and beyond. Equations are mathematical statements that hold true for specific values of the variable(s), while identities are true for all permissible values. This article explores their definitions, methods of solving equations, application of identities for simplification, and key exam techniques aligned with the Edexcel specification. We will traverse from linear equations to trigonometric identities, equipping you with clarity and confidence.
在A-Level数学中，掌握方程与恒等式是解决代数、三角等问题的基石。方程是指对变量的某些特定值成立的等式，而恒等式则是对变量所有允许值都成立的恒等关系。本文将深入探讨它们的定义、解方程的方法、恒等式的应用以及针对Edexcel大纲的应试技巧，帮助你从线性方程到三角恒等式建立起清晰的理解与自信。
1. Definitions: Equations vs Identities | 定义：方程与恒等式
an equation with the solution x = 2. In contrast, an identity is an equality that holds for all values of the variable for which both sides are defined. The symbol '≡' is often used to stress an identity, such as 2(x + 1) + 1 ≡ xpressions or prove other relationships, whereas equations require finding specific roots.
方程是表明两个表达式相等的命题，但只在未知数取特定值时才成立。例如，2x + 3 = 7 是一个方程，解为 x = 2。相反，恒等式则是对变量所有可能取值都恒成立的等式，常用'≡'符号强调，比如 2(x + 1) + 1 ≡ 2x + 3。区分二者至关重要，因为恒等式可用于化简表达式或证明其他关系，而方程则需要求解具体数值。
In Edexcel exams, you may be explicitly asked to determine whether a given statement is an equation or an identity. A quick check is to attempt to simplify both sides; if they reduce to the same polynomial form, it is an identity. If not, it is an equation with limited solutions.
在Edexcel的考试中，你可能会被明确要求判断一个式子是否为恒等式。快速检验的方法是尝试化简两边：如果化简后得到相同的多项式，那就是恒等式；否则便是一个只能对有限解成立的方程。
2. Solving Linear Equations | 解线性方程
xpand brackets, collect like terms, and then apply addition/subtraction and multiplication/division. For instance, to solve 3(2x – 4) + 5 = 2x + 7, first expand to get 6x – 12 + 5 = 2x + 7, simplify to 6x – 7 = 2x + 7, then subtract 2x and add 7 to both sides, yielding 4x = 14, so x = 3.5.
线性方程仅含一次方的变量。求解的一般步骤是通过逆运算分离未知数：展开括号、合并同类项，然后运用加减和乘除。例如，解 3(2x – 4) + 5 = 2x + 7，首先展开得 6x – 12 + 5 = 2x + 7，化简为 6x – 7 = 2x + 7，再两边减 2x 并加 7，得 4x = 14，于是 x = 3.5。
Always check your solution by substituting it back into the original equation. This step verifies correctness and avoids simple algebraic errors. Linear equations with fractions can be simplified by multiplying through by the common denominator.
始终通过将解代回原方程来检验答案，这能验证正确性并避免简单代数错误。带有分数的线性方程可以通过乘以公分母来简化。
3. Solving Quadratic Equations | 解二次方程
formula. Factorisation works when the quadratic can be expressed as (px + q)(rx + s) = 0, giving solutions x = –q/p or x = –s/r. For example, x² – 5x + 6 = 0 factorises to (x – 2)(x – 3) = 0, hence x = 2 or 3.
二次方程具有形式 ax² + bx + c = 0，且 a ≠ 0。主要采用三种方法：因式分解、配方法和求根公式。当二次式可分解为 (px + q)(rx + s) = 0 时，因式分解可得出解 x = –q/p 或 x = –s/r。例如，x² – 5x + 6 = 0 可分解为 (x – 2)(x – 3) = 0，因此 x = 2 或 3。
If factorisation proves difficult, the quadratic formula is reliable:
x = ( –b ± √b² – 4ac ) / (2a)
The discriminant Δ = b² – writes the quadratic in vertex form a(x + p)² + q, which is particularly useful for sketching graphs and finding maxima/minima.
如果因式分解较为困难，求根公式是最可靠的方法：
x = ( –b ± √b² – 4ac ) / (2a)
判别式 Δ = b² – 4ac 指示根的性质：Δ > 0 有两个
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