Essential Mathematical Formulas (Basic) | 数学常用公式汇总（基础篇）

📚 Essential Mathematical Formulas (Basic) | 数学常用公式汇总（基础篇）

Mastering basic mathematical formulas is the first step toward success in any math course. This article compiles essential formulas from algebra, geometry, trigonometry, and coordinate geometry to help you build a strong foundation. Whether you are preparing for an exam or refreshing your knowledge, keep these formulas handy.

掌握基础数学公式是学好数学的第一步。本文汇总了代数、几何、三角与坐标几何中常用的公式，帮助你打下扎实的基础。无论你在备考还是巩固知识，都可以随时查阅这些公式。

1. Algebraic Identities | 代数恒等式

Algebraic identities are equations that hold true for all values of the variables. They simplify expressions and solve equations efficiently.

代数恒等式对所有变量取值都成立，可以用于简化表达式和高效解方程。

Square of a sum: (a + b)² = a² + 2ab + b²

和的平方：(a + b)² = a² + 2ab + b²

Square of a difference: (a − b)² = a² − 2ab + b²

差的平方：(a − b)² = a² − 2ab + b²

Difference of squares: a² − b² = (a − b)(a + b)

平方差：a² − b² = (a − b)(a + b)

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)

立方和：a³ + b³ = (a + b)(a² − ab + b²)

Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)

立方差：a³ − b³ = (a − b)(a² + ab + b²)

2. Quadratic Formula | 二次方程求根公式

For a quadratic equation in the standard form ax² + bx + c = 0 (a ≠ 0), the solutions are given by the quadratic formula:

对于标准形式 ax² + bx + c = 0（a ≠ 0）的二次方程，其解可通过求根公式得出：

x = [−b ± √(b² − 4ac)] / (2a)

The expression under the square root, D = b² − 4ac, is called the discriminant. It determines the nature of the roots: if D > 0, two distinct real roots; if D = 0, one repeated real root; if D < 0, no real roots.

根号内的表达式 D = b² − 4ac 称为判别式，它决定了根的性质：若 D > 0，有两个不等实根；若 D = 0，有一个重实根；若 D < 0，无实根。

3. Laws of Indices | 指数定律

Indices (exponents) follow a set of rules that allow us to manipulate powers efficiently.

指数（幂）遵循一组运算规律，使我们可以高效处理幂的运算。

Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (where a ≠ 0)

乘法：aᵐ × aⁿ = aᵐ⁺ⁿ（a ≠ 0）

Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power of a power: (aᵐ)ⁿ = aᵐⁿ

幂的幂：(aᵐ)ⁿ = aᵐⁿ

Power of a product: (ab)ⁿ = aⁿ bⁿ

积的乘方：(ab)ⁿ = aⁿ bⁿ

Power of a quotient: (a/b)ⁿ = aⁿ / bⁿ (b ≠ 0)

商的乘方：(a/b)ⁿ = aⁿ / bⁿ（b ≠ 0）

Zero exponent: a⁰ = 1 (a

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