📚 Mastering Equations and Inequalities for Edexcel A-Level Mathematics | 掌握方程与不等式：Edexcel A-Level数学必备
Equations and inequalities form the foundation of A-Level Mathematics. Mastering these skills is essential for solving problems across pure mathematics, mechanics, and statistics. This revision guide covers all key topics from linear equations to modulus inequalities, aligned with the Edexcel specification.
方程与不等式是A-Level数学的基础。掌握这些技能对于解决纯数学、力学和统计问题至关重要。本复习指南涵盖从线性方程到绝对值不等式的所有关键主题，符合Edexcel考试大纲。
1. Solving Linear Equations | 解线性方程
btract 2x from both sides to get x + 5 = −4, then subtract 5 to obtain x = −9. The same principle applies when the equation contains fractions: multiply through by the lowest common denominator to clear denominators, then solve as usual. Always check your solution by substituting it back into the original equation.
线性方程涉及一次幂的变量，可以通过隔离未知数来求解。例如，解 3x + 5 = 2x − 4，从两边减去 2x 得到 x + 5 = −4，然后减去 5 得到 x = −9。当方程含有分数时，同样可以两边乘以最小公分母消去分母，然后照常求解。始终将解代入原方程进行检验。
2. Solving Quadratic Equations | 解二次方程
uare, and using the quadratic formula. When factorising a monic quadratic x² + bx + c, find two numbers that multiply to c and add to b. For x² − 5x + 6 = 0, the numbers are −2 and −3, giving (x − 2)(x − 3) = 0, so x = 2 or x = 3. If the coefficient of x² is not 1, use the 'ac method' or trial and error.
二次方程的形式为 ax² + bx + c = 0，其中 a ≠ 0。解二次方程的主要三种方法是因式分解、配方法和使用二次公式。在因式分解首项系数为1的二次式 x² + bx + c 时，找到两个数，使得它们的乘积为 c，和为 b。对于 x² − 5x + 6 = 0，这两个数是 −2 和 −3，得到 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。如果 x² 的系数不为1，则使用“ac方法”或试错法。
3. Completing the Square and the Quadratic Formula | 配方法与二次公式
et. For example, x² + 6x + 5 becomes (x + 3)² − 4. The quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), works for any quadratic when factorisation is difficult. The discriminant Δ = b² − 4ac determines the nature of the roots:
配方法将 ax² + bx + c 重写为 a(x + p)² + q 的形式。首先从含 x 的项中提取 a，然后在括号内加上并减去 (b/2a)²。例如，x² + 6x + 5 变为 (x + 3)² − 4。二次公式 x = (−b ± √b² − 4ac) / (2a) 适用于任何难以因式分解的二次方程。判别式 Δ = b² − 4ac 决定了根的性质：
| b² − 4ac | Nature of roots | 根的性质 |
|---|---|---|
| Δ > 0 | Two distinct real roots | 两个不等实根 |
| Δ = 0 | One repeated real root | 一个重实根 |
| Δ < 0 | No real roots (two complex roots) | 无实根（两个复根） |
4. Simultaneous Equations | 联立方程组
ctive. When one equation is linear and the other is quadratic, substitution usually works best. For example, given y = 2x + 1 and x² + y² = 10, replace y in the second equation: x² + (2x + 1)² = 10. Expanding gives 5x² + 4x − 9 = 0, which factorises to (5x + 9)(x − 1) = 0, so x = 1 or x = −9/5. Find the corresponding y values and check both pairs satisfy the original equations.
联立方程组包含两个或多个具有多个变量的方程。对于线性方程组，消元法或代入法同样有效。当一个方程为线性、另一个为二次时，代入法通常最有效。例如，给定 y = 2x + 1 和 x² + y² = 10，将第二个方程中的 y 替换：x² + (2x + 1)² = 10。展开得 5x² + 4x − 9 = 0，因式分解为 (5x + 9)(x − 1) = 0，因此 x = 1 或 x = −9/5。求出相应的 y 值，并检验两组解是否满足原方程。
5. Inequalities and Interval Notation | 不等式与区间表示法
An inequality states a relationship using symbols <, >, ≤, ≥. Solutions are expressed on a number line or with interval notation: parentheses ( ) for strict inequalities, square brackets [ ] for inclusive bounds. For example, x > 3 is written as (3, ∞), while −2 ≤ x < 5 is [−2, 5). When multiplying or dividing an inequality by a negative number, you must reverse the inequality sign.
不等式使用 <、>、≤、≥ 等符号表示关系。解集用数轴或区间表示法表示：严格不等式使用圆括号 ( )，包含边界的使用方括号 [ ]。例如，x > 3 写作 (3, ∞)，而 −2 ≤ x < 5 写作 [−2, 5)。在乘以或除以负数时，必须反转不等号的方向。
6. Linear Inequalities | 线性不等式
Solving linear inequalities follows the same algebraic steps as linear equations, but the solution is an interval rather than a single value. Consider 4 − 3x < 2x + 9. Collect x terms on one side: −3x − 2x < 9 − 4, so −5x < 5. Dividing by −5 reverses the sign, giving x > −1. Thus the solution set is (−1, ∞). Double-ended inequalities of the form a < f(x) < b can be solved by breaking them into two separate inequalities or manipulating all three parts simultaneously, ensuring the same operation is applied to each part.
解线性不等式的步骤与线性方程相同，但解是一个区间而非单一数值。例如 4 − 3x < 2x + 9。将含 x 的项移项：−3x − 2x < 9 − 4，得 −5x < 5。除以 −5 反转不等号，得到 x > −1。因此解集为 (−1
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