12 Solving Equations | 12 解方程

📚 12 Solving Equations | 12 解方程

Solving equations is a core skill in A-Level mathematics. An equation states that two expressions are equal, and solving it means finding all values of the variable that make the statement true. This article reviews the main types of equations you need to solve and the most efficient methods for each.

解方程是 A-Level 数学的核心技能。方程表示两个表达式相等，解方程就是找出使该等式成立的所有变量值。本文回顾你需要掌握的主要方程类型及每种类型最有效的方法。

1. Linear Equations | 线性方程

Linear equations have no powers higher than 1. The simplest form is ax + b = c. Use addition, subtraction, multiplication and division to isolate the variable.

线性方程不含高于一次幂的项。最简单的形式是 ax + b = c。通过加减乘除隔离变量。

For example, solve 3x − 7 = 11. Add 7 to both sides: 3x = 18. Divide by 3: x = 6. Always check by substitution.

例如，解 3x − 7 = 11。两边加 7 得 3x = 18。除以 3 得 x = 6。务必代入验算。

When the unknown appears on both sides, collect the variable terms on one side and the constants on the other. Solve 5x + 3 = 2x + 12: subtract 2x from both sides to get 3x + 3 = 12, then 3x = 9, so x = 3.

当未知数出现在等式两边时，将含未知数的项集中到一边，常数移到另一边。解 5x + 3 = 2x + 12：两边减去 2x 得 3x + 3 = 12，则 3x = 9，所以 x = 3。

2. Quadratic Equations | 二次方程

A quadratic equation has the form ax² + bx + c = 0. Three standard methods are factorisation, completing the square and the quadratic formula.

二次方程的形式为 ax² + bx + c = 0。三种标准方法是因式分解、配方法和求根公式。

Factorisation works when the equation can be written as (x + p)(x + q) = 0. For x² + 5x + 6 = 0, we have (x + 2)(x + 3) = 0, so x = −2 or x = −3.

当方程可写成 (x + p)(x + q) = 0 时可用因式分解。对于 x² + 5x + 6 = 0，有 (x + 2)(x + 3) = 0，所以 x = −2 或 x = −3。

Completing the square rearranges ax² + bx + c = 0 into a(x + h)² + k = 0. For x² + 6x + 5 = 0, complete the square to get (x + 3)² = 4, then x + 3 = ±2, giving x = −1 or x = −5.

配方法将 ax² + bx + c = 0 化为 a(x + h)² + k = 0。对于 x² + 6x + 5 = 0，配方得 (x + 3)² = 4，则 x + 3 = ±2，得 x = −1 或 x = −5。

For any quadratic, the quadratic formula gives:

对于任意二次方程，求根公式为：

x = (−b ± √(b² − 4ac)) / (2a)

The discriminant Δ = b² − 4ac determines the nature of the roots. If Δ > 0 there are two distinct real roots; if Δ = 0 one repeated real root; if Δ < 0 no real roots.

判别式 Δ = b² − 4ac 决定根的性质。若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。

3. Simultaneous Linear Equations | 线性联立方程

Two linear equations in two unknowns can be solved by elimination or substitution. For elimination, make the coefficients of one variable equal, then add or subtract the equations.

两个含两个未知数的线性方程可用消元法或代入法求解。消元法通过使某个变量的系数相同，再相加或相减两个方程。

Example: 2x + y = 7 and 3x − y = 3. Adding gives 5x = 10, so x = 2. Substituting into the first equation gives 4 + y = 7, so y = 3.

例：2x + y = 7 与 3x − y = 3。相加得 5x = 10，所以 x = 2。代入第一个方程得 4 + y = 7，所以 y = 3。

Substitution involves rearranging one equation for a variable and substituting into the other. It is particularly useful when one equation is already in the form y = mx + c.

代入法是将一个方程变形为某个变量表示后代入另一个方程。当一个方程已经是 y = mx + c 的形式时尤为方便。

4. Simultaneous Equations with a Quadratic | 含二次的联立方程

When one equation is linear and

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