负数乘方的奇偶规律$$(-2)^3 = -8,\ (-2)^4 = 16,\ (-1)^{2n} = 1,\ (-1)^{2n+1} = -1$$

一元一次方程解法$$\frac{x}{2} - \frac{x+1}{3} = 1 \implies 3x - 2(x+1) = 6 \implies x = 8$$

三角形内角和与外角定理$$\angle A + \angle B + \angle C = 180^{\circ},\quad \text{外角} = \angle A + \angle B$$

二元一次方程组求解(代入消元)$$\begin{cases} 2x + 3y = 8 \\ x - y = 1 \end{cases} \implies x = \frac{11}{5},\ y = \frac{6}{5}$$

实数常用值速记$$\sqrt{2} \approx 1.414,\ \sqrt{3} \approx 1.732,\ \sqrt[3]{-8} = -2,\ \pi \approx 3.14159$$

乘法公式(因式分解的逆)$$(a+b)(a-b) = a^2 - b^2,\quad (a \pm b)^2 = a^2 \pm 2ab + b^2$$

分式方程求解与增根检验$$\frac{1}{x-1} + \frac{2}{x+1} = 1 \implies x = 3,\ \text{检验 } x \neq \pm 1$$

反比例函数的几何意义$$y = \frac{k}{x}\ (k \neq 0),\quad S_{\text{矩形}} = |k|,\quad S_{\text{三角形}} = \frac{|k|}{2}$$

勾股定理与斜边求解$$a^2 + b^2 = c^2,\quad c = \sqrt{a^2 + b^2}$$

二次根式性质速记$$\sqrt{a^2} = |a|,\quad (\sqrt{a})^2 = a\ (a \geq 0),\quad \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

一元二次方程求根公式 + 判别式 + 韦达定理$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\quad \Delta = b^2 - 4ac,\quad x_1 + x_2 = -\frac{b}{a},\ x_1 x_2 = \frac{c}{a}$$

圆的周长 / 面积 / 弧长 / 扇形面积$$C = 2\pi r,\quad S = \pi r^2,\quad l = \frac{n\pi r}{180},\quad S_{\text{扇}} = \frac{n\pi r^2}{360}$$

古典概型概率公式$$P(A) = \frac{m}{n},\quad m = \text{事件 A 包含结果数},\ n = \text{总结果数}$$

二次函数顶点式 + 对称轴 + 顶点坐标$$y = a(x-h)^2 + k,\quad \text{对称轴 } x = -\frac{b}{2a},\quad \text{顶点 } \left(-\frac{b}{2a},\ \frac{4ac-b^2}{4a}\right)$$

相似三角形对应边比 = 相似比,面积比 = 相似比²$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k,\quad \frac{S_{\triangle ABC}}{S_{\triangle DEF}} = k^2$$

特殊角三角函数值 + 平方关系$$\sin 30^{\circ} = \frac{1}{2},\ \cos 30^{\circ} = \frac{\sqrt{3}}{2},\ \tan 30^{\circ} = \frac{\sqrt{3}}{3},\quad \sin^2\theta + \cos^2\theta = 1$$