📚 1A Basic Techniques | 1A 基本技巧
In IB Mathematics, mastering basic algebraic techniques is essential for success in both Analysis and Approaches (AA) and Applications and Interpretation (AI). This section reviews the fundamental skills you will rely on throughout the course, from simplifying expressions to solving equations, working with exponents, and using your GDC effectively. A solid grasp of these techniques will make more advanced topics much easier to handle.
在 IB 数学中，掌握基本的代数技巧对于分析数学 (AA) 和应用数学 (AI) 的成功至关重要。本节回顾了你在整个课程中依赖的基本技能，从化简表达式到解方程、运用指数法则和有效使用图形计算器。扎实掌握这些技巧将使处理更高级的专题变得容易得多。
1. Expanding Brackets | 展开括号
with two binomials, such as (x + a)(x + b), use the FOIL method: multiply First, Outer, Inner, Last terms, then simplify. For a perfect square like (x + a)², remember the pattern: x² + 2ax + a². The difference of two squares, (x + a)(x – a), simplifies to x² – a².
展开括号是指通过将括号内的每一项与外面的项相乘来去掉括号。对于单个括号，遵循分配律：a(b + c) = ab + ac。处理两个二项式时，例如 (x + a)(x + b)，使用 FOIL 方法：先乘首项、外项、内项、末项，然后化简。对于完全平方如 (x + a)²，记住模式：x² + 2ax + a²。平方差 (x + a)(x – a) 简化为 x² – a²。
y or use a systematic approach such as a grid method. Always watch out for negative signs: –(x – y) = –x + y. Practice with expressions like 3x(2x² – 5x + 4) and (2x – 3)(x + 1)² to build fluency.
当展开三项或更多项时，重复应用分配律或使用网格法等系统方法。务必注意负号：–(x – y) = –x + y。通过练习如 3x(2x² – 5x + 4) 和 (2x – 3)(x + 1)² 等表达式来提高熟练度。
2. Factorising | 因式分解
the coefficient of x² is not 1, use methods like splitting the middle term or the AC method. For 2x² + 7x + 3, the product AC = 6, so split 7x into 6x + x, then factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
因式分解是展开的逆过程。目标是将表达式写成其因子的乘积形式。首先寻找公因子：例如 6x² + 9x = 3x(2x + 3)。对于形如 x² + bx + c 的二次式，找出两个数使得它们的积为 c 且和为 b。例如 x² + 5x + 6 = (x + 2)(x + 3)。当 x² 的系数不为 1 时，使用拆中项法或 AC 法。对于 2x² + 7x + 3，乘积 AC = 6，因此将 7x 拆分为 6x + x，然后分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
r factorisation by expanding mentally. Special forms include the difference of squares (x² – a² = (x – to factorise expressions with multiple variables, like xy + 2x + 3y + 6 = (x + 3)(y + 2).
始终通过心算展开来检验因式分解。特殊形式包括平方差 (x² – a² = (x – a)(x + a))、完全平方三项式 (x² ± 2ax + a² = (x ± a)²) 以及立方和/差（尽管出现频率较低）。还要能分解含多个变量的表达式，如 xy + 2x + 3y + 6 = (x + 3)(y + 2)。
3. Solving Linear Equations | 解线性方程
sides: add/subtract terms, then multiply/divide by coefficients. For example, 3x + 5 = 2x – 7 → x = –12. Always check your solution by substituting back into the original equation.
线性方程的形式为 ax + b = cx + d。解法是合并同类项并分离变量。在等式两边执行相同的运算：先加减项，再乘除以系数。例如 3x + 5 = 2x – 7 → x = –12。始终将解代回原方程进行验证。
lear denominators. For example, (x/2) + (1/3) = (x/4) → multiply by 12: 6x + 4 = 3x → 3x = –4 → x = – of restrictions such as denominators not being zero. Equations with absolute values, like |x – 2| = 5, lead to two cases: x – 2 = 5 or x – 2 = –5, giving x = 7 or x = –3.
如果方程含有分数，两边乘以最小公分母以去分母。例如 (x/2) + (1/3) = (x/4) → 两边乘以 12：6x + 4 = 3x → 3x = –4 → x = –4/3。注意分母不能为零等限制条件。含绝对值的方程如 |x – 2| = 5，分为两种情况：x – 2 = 5 或 x – 2 = –5，得 x = 7 或 x = –3。
4. Solving Quadratic Equations | 解二次方程
tic formula. If the expression can be factorised, set each factor to zero. For example, x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 or x = 3.
二次方程 (ax² + bx + c = 0, a ≠ 0) 可以通过因式分解、配方法或二次公式求解。如果表达式能够因式分解，则令每个因式为零。例如 x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 或 x = 3。
The quadratic formula x = [–b ± √b² – 4ac] / (2a) works for any quadratic. The discriminant Δ = b² – < 0, no real roots (two complex). Remember to simplify surds in exact answers, e.g., leave √8 as 2√2. Completing the square is especially useful for deriving the vertex form of a quadratic function and will be detailed later.
二次公式 x = [–b ± √b² – 4ac] / (2a) 对所有二次方程都适用。判别式 Δ = b² – 4ac 指示根的性质：若 Δ > 0，有两个不等实根；Δ = 0，有一个重实根；Δ < 0，无实根（有两个复数根）。在精确答案中要化简根式，例如将 √8 写成 2√2。配方法对于推导二次函数的顶点式特别有用，稍后将详述。
e equation to zero before solving. Sometimes you may need to rearrange: 2x² = 5x + 3 → 2x² – 5x – 3 = 0. When the equation contains fractions or roots, simplify first.
记住在求解前先将方程设为零。有时需要移项：2x² = 5x + 3 → 2x² – 5x – 3 = 0。当方程含有分数或根号时，先化简。
5. Laws of Exponents | 指数定律
Exponents (indices) follow a set of logical rules. For any real numbers a and b (with a, b ≠ 0 where necessary) and integers m, n:
指数（幂）遵循一套逻辑规则。对于任意实数 a 和 b（必要时 a, b ≠ 0）及整数 m, n：
- Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ
- Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a power: (aᵐ)ⁿ = aᵐⁿ
- Power of a product: (ab)ⁿ = aⁿ bⁿ
- Zero exponent: a⁰ = 1 (a ≠ 0)
- Negative exponent: a⁻ⁿ = 1/aⁿ
- Fractional exponents: a1/n = ⁿ√a , am/n = ⁿ√aᵐ
- 乘法：aᵐ × aⁿ = aᵐ⁺ⁿ
- 除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- 幂的幂：(aᵐ)ⁿ = aᵐⁿ
- 积的幂：(ab)ⁿ = aⁿ bⁿ
- 零指数：a⁰ = 1 (a ≠ 0)
- 负指数：a⁻ⁿ = 1/aⁿ
- 分数指数：a1/n = ⁿ√a ， am/n = ⁿ√aᵐ
th negative signs and always express final answers with positive exponents where possible.
应用这些规则化简诸如 (2x²y⁻³)⁴ ÷ (x⁻⁵y²) = 16 x⁸⁺⁵ y⁻¹²⁻² = 16 x¹³ y⁻¹⁴ 的表达式。注意负号，并尽可能用正指数表示最终答案。
6. Surds and Radicals | 根式与无理数
Surds are irrational roots, such as √2 or ∛5, that cannot be written as exact decimals. Simplifying surds involves expressing them in the form a√b where b has no square factors. For example, √50 = √25×2 = 5√2. Operations with surds follow the same rules as algebraic expressions: a√b + c√b = (a+c)√b, and √a × √b = √ab. Rationalising the denominator removes surds from the bottom of a fraction: multiply numerator and denominator by the conjugate. For instance, 1/(√2 – 1) = (√2 + 1)/((√2–1)(√2+1)) = √2 + 1.
根式是无理根，如 √2 或 ∛5，无法写成精确小数。化简根式意味着将其写成 a√b 的形式，其中 b 不含平方因子。例如 √50 = √25×2 = 5√2。根式的运算遵循与代数式相同的规则：a√b + c√b = (a+c)√b，且 √a × √b = √ab。分母有理化可以消去分数下方的根号：分子分母同时乘以共轭式。例如 1/(√2 – 1) = (√2 + 1)/((√2–1)(√2+1)) = √2 + 1。
When solving equations with surds, square both sides carefully, checking for extraneous solutions. The key properties √a² = |a| and (√a)² = a (for a ≥ 0) are essential in many IB problems.
当求解含有根式的方程时，对方程两边平方需谨慎，并验根排除增根。性质 √a² = |a| 和 (√a)² = a (对于 a ≥ 0) 在许多 IB 问题中至关重要。
7. Algebraic Fractions | 代数分式
es factorising numerator and denominator, then cancelling common factors. For example, (x² – 4)/(x² + 2x) = ((x–2)(x+2))/(x(x+2)) = (x–2)/x, provided x ≠ –2, 0. To add or subtract fractions, find a common denominator: 1/(x–1) + 2/(x+2) = (x+2 + 2(x–1))/((x–1)(x+2)) = (3x)/((x– ltiplication and division follow the usual rules: multiply by the reciprocal for division.
化简代数分式需要先将分子分母因式分解，然后约去公因子。例如 (x² – 4)/(x² + 2x) = ((x–2)(x+2))/(x(x+2)) = (x–2)/x，其中 x ≠ –2, 0。加减分式时要找到公分母：1/(x–1) + 2/(x+2) = (x+2 + 2(x–1))/((x–1)(x+2)) = (3x)/((x–1)(x+2))。乘法和除法遵循通常的规则：除法时乘以倒数。
Always state restrictions on the variable (values that make a denominator zero). In IB exams, these are often part of the mark scheme. When solving rational equations, multiply through by the common denominator, solve, and check that none of the solutions are restricted.
始终说明变量的限制条件（使分母为零的值）。在 IB 考试中，这通常是评分方案的一部分。当解有理方程时，两边乘以公分母，求解，并检查所得解是否不在限制范围之内。
8. Completing the Square | 配方法
ng the square transforms a quadratic expression of the form ax² + bx + c into the form a(x – + k. For a monic quadratic x² + bx + c, add and subtract (b/2)²: x² + bx + c = (x + b/2)² – (b/2)² + c. For example, x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – of a parabola, solve quadratic equations without a formula, and simplify integrals later.
配方法将形如 ax² + bx + c 的二次表达式转化为 a(x – h)² + k 的形式。对于首项系数为 1 的二次式 x² + bx + c，加上并减去 (b/2)²：x² + bx + c = (x + b/2)² – (b/2)² + c。例如 x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4。如果 a ≠ 1，先对 x² 和 x 项提取因子 a。这一技巧用于求抛物线的顶点、不用公式解二次方程，以及后续简化积分。
To solve x² + 6x + 5 = 0 by completing the square, write (x + 3)² – 4 = 0 → (x + 3)² = 4 → x + 3 = ±2 → x = –1 or x = – ves the exact same answers as factorisation. Always remember the constant term adjustment.
要用配方法解 x² + 6x + 5 = 0，写成 (x + 3)² – 4 = 0 → (x + 3)² = 4 → x + 3 = ±2 → x = –1 或 x = –5。此方法得出与因式分解完全相同的解。永远记住调整常数项。
9. Simultaneous Equations | 联立方程
elimination. For linear equations, elimination is often faster: e.g., 3x + 2y = 16 and 2x – y = 6. Multiply the second by 2 → 4x – quation and plugging it into the other, especially useful when one equation is non-linear.
联立方程是寻找同时满足两个或更多方程的值的组合。方法包括代入法和消元法。对于线性方程，消元法通常更快：例如 3x + 2y = 16 和 2x – y = 6。将第二个乘以 2 → 4x – 2y = 12，与第一个相加 → 7x = 28 → x = 4，然后 y = 2。代入法涉及从一个方程中分离一个变量并代入另一个方程，当其中一个方程为非线性时尤其有用。
ratic equation in one variable. For instance, x + y = 5 and x² + y² = 25: substitute y = 5 – x → x² + (5 – x)² = 25 → 2x² – 10x = 0 → x(x – al interpretation is the intersection points of curves, which can be verified on your GDC.
当其中一个方程为二次时，代入会导出一个关于一个变量的二次方程。例如 x + y = 5 和 x² + y² = 25：代入 y = 5 – x → x² + (5 – x)² = 25 → 2x² – 10x = 0 → x(x – 5) = 0，得到 (0,5) 和 (5,0)。始终将解代入两个原方程检验。几何解释是曲线交点，可在图形计算器上验证。
10. Inequalities | 不等式
ut multiplying or dividing by a negative number reverses the inequality sign. For example, –2x > 6 → x < –3. Quadratic inequalities require a sign diagram or graph. For x² – 3x – 4 ≤ 0, factorise to (x– ≤ 0, determine where the product is ≤ 0 by testing intervals. The critical values are x = –1 and x = 4. Testing gives –1 ≤ x ≤ 4. Remember to use solid dots for ≤ or ≥, open dots for < or > on a number line.
解线性不等式与方程相似，但乘以或除以负数时要反转不等号方向。例如 –2x > 6 → x < –3。二次不等式需要符号表格或图形。对于 x² – 3x – 4 ≤ 0，因式分解得 (x–4)(x+1) ≤ 0，通过测试区间确定乘积 ≤ 0 的 x 值。临界值为 x = –1 和 x = 4。测试得 –1 ≤ x ≤ 4。记住在数轴上 ≤ 或 ≥ 用实心圆点，< 或 > 用空心圆点。
Systems of linear inequalities define regions in the coordinate plane. Shade the feasible region that satisfies all inequalities. In IB, you might be asked to find the maximum or minimum of an objective function over this region, an introduction to linear programming. Always clearly label boundary lines and indicate whether they are included or not.
线性不等式组在坐标平面中定义区域。涂出满足所有不等式的可行域。在 IB 中，你可能被要求在可行域上求目标函数的最大值或最小值，这是线性规划的入门。务必清楚地标记边界线，并表明是否包含边界。
11. Basic Functions and Notation | 基本函数与符号
ke denominators or even roots). Range is the set of all possible output values. For f(x) = √x and range y ≥ 0. For rational functions, exclude x-values that make the denominator zero.
函数 f 将定义域中的每个输入 x 映射到值域中恰好一个输出 f(x)。常见记法：f(x) = x² + 1。垂直线检验可判断一个图形是否表示函数。定义域指所有允许的 x 值（通常为所有实数，除非有分母或偶次根号等限制）。值域是所有可能的输出值的集合。对于 f(x) = √x，定义域为 x ≥ 0，值域为 y ≥ 0。对于有理函数，排除使分母为零的 x 值。
ve for y. Not all functions have inverses unless they are one-to-one (pass horizontal line test). Restricting the domain often creates an invertible function.
复合函数如 f(g(x)) 表示先应用 g 再应用 f。f(g(x)) 的定义域需考虑 g 的定义域和 f 的定义域。反函数 f⁻¹(x) 逆反操作，使得 f(f⁻¹(x)) = x。求反函数时，在方程 y = f(x) 中交换 x 和 y 并解出 y。并非所有函数都有反函数，除非是一对一的（通过水平线检验）。限定定义域常可构造出可逆函数。
12. Using the GDC Effectively | 高效使用图形计算器
The graphical display calculator (GDC) is a powerful tool in IB Mathematics. Learn to graph functions and adjust the viewing window (ZOOM) to see key features. Find roots (x-intercepts), maxima/minima, and intersections of two graphs quickly using the CALC menu. In the equation solver, you can enter an equation and it returns a numerical solution, which is especially handy for complex equations that cannot be solved algebraically. Always check if an exact answer is required; the GDC gives decimal approximations unless you set it to display exact forms.
图形计算器 (GDC) 是 IB 数学中的强大工具。学会绘制函数图并调整视窗 (ZOOM) 以观察关键特征。使用 CALC 菜单快速求根（x 截距）、最大值/最小值以及两条图线的交点。在方程求解器中，你可以输入方程并得到数值解，这对于无法用代数方法求解的复杂方程尤为方便。始终检查是否需要精确答案；除设置显示精确形式外，GDC 给出的是小数近似值。
Your GDC also handles matrices, statistics, probability distributions, and calculus operations like numerical derivatives and definite integrals. Familiarize yourself with its memory and list functions for efficient data handling. Practice using it for verifying algebraic work, exploring transformations, and solving systems of equations graphically. However, never rely on GDC as a substitute for showing algebraic reasoning in exam questions that require it.
你的 GDC 还能处理矩阵、统计、概率分布，以及数值导数和定积分等微积分运算。熟悉其存储和列表功能以便高效处理数据。练习使用它验证代数运算、探索变换以及图形化求解方程组。然而，在考试题要求展示代数推理时，绝不能依赖 GDC 替代真正的代数过程。
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