# Formulas

The document contains many mathematical formulas across various topics: 1) Formulas for solving equations like quadratic, logarithmic, and trigonometric equations. 2) Formulas for limits, derivatives, integrals, and operations involving exponents, logarithms, and radicals. 3) Set notation formulas defining sets of real numbers based on conditions.

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### Formulas

• 1. Formulas: 1 2 𝑥: ̂ ±𝑏− √𝜋∗∝ 𝑃: 5 log3(𝑥+ √ ̂ 𝛽) (2𝑥𝑦)2 ∗(3𝑎𝑏+3𝑥)3 3 √ 𝑥 3 ∗𝑦 2 = 𝑅𝑇 − 𝑣−𝑏 𝑐 𝑎∗𝑑+𝑏∗𝑐 𝑥 3 −𝑦2 √2 𝑎𝑛∗ 𝑏 𝑚 = 𝑎 𝑛+𝑚 𝑎 𝑛 /𝑏 𝑐 + 𝑏= 𝑏 𝑚 1 𝑎 𝑚 (𝑎2 +𝑏 2 +𝑥 2 +𝑦 2 )∗(𝑥 3 +𝑏 3 ) 𝑎3 +𝑏2 𝑥1= 𝑐 − 𝑏= 𝑏 𝑚 𝑎− 𝑛 = 𝑚 √3𝑥−2𝑦 3 −23 (𝑥 𝑎 −𝑦 𝑏 )∗(32 +2𝑥−𝑦 3 ) 𝑎 𝑏 𝑛 𝑎3 = = 2√ 𝑎+𝑐 𝑎𝑛 = √𝑎 𝑛 23 ∗ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 𝑎 𝑎−𝑛 = 𝑏∗𝑑 𝑣2 1 2 𝑦2 𝑎 𝑥 + +𝑏 𝑥 22 ∗𝑎2 ∗𝑏 2 ̅̅̅// ⃑⃑⃑⃑ = ̂ ∩ 𝑒𝑏 𝑑: {𝑒} ̂ 𝑎𝑏 𝑐𝑑 𝑎𝑏 + 𝑑= 𝑏 √ 5𝑥 2 +3𝑦2 −𝑎3 −𝑏3 ∀𝑥 ∈ ℜ: ∃𝑦 ∈/x≥ 𝑜 ⋏ 𝑦 ∈ 𝑥 𝑎 𝑎 −𝑏+√𝑏2 4𝑎𝑐 2 𝑎 𝑎−𝑐 𝑏 𝑥2= −𝑏−√𝑏 2 4𝑎𝑐 𝑏 𝑎∗ 𝑐 = 2𝑎 𝑎∗𝑐+𝑏 𝑐 1 𝑛 √𝑎 𝑚 = 𝑎 𝑛−𝑚 𝜋 𝐷 = {𝑥 ∈ ℝ/∃𝑓(𝑥)} 𝐷 = ℝ − {(2𝐾 + 1) ∗ 2 ; 𝐾 ∈ 𝑍} 𝐷 = ℝ − {𝐾𝜋; ∈ 𝑍} 𝑥 > 𝑎 ⟹ 𝑓(𝑥) ≤ 𝑓(𝑎) ∫ 𝑑𝑥 = 𝑥 + 𝑐 lim 𝑓(𝑥) = ∞ ⟺ ∀𝐾 ∈ ℝ+ ∃𝛿(𝐾) > 0/0 < |𝑥 − 𝑥0 | < 𝛿 ⟹ 𝑓(𝑥) > 𝐾 𝑥→𝑎 𝑛 𝑎 𝑛+1 − 𝑏 𝑛−1 = (𝑎 + 𝑏) ∑ 𝑘=0 𝑎 𝑘 𝑏 𝑛−𝑘 log 𝑎 𝑥 = 𝑦 ⇒ 𝑎 𝑦 = 𝑥 log 𝑎 𝑛 log 𝑎 𝑛 𝑑𝑥 = ∫ 𝐶(𝑥)𝑑𝑥 + ∫ log log 𝑎 𝑥 = log 𝑏 ( √ 𝑥) = 𝑛 log 𝑎 𝑥 𝑛 𝑃(𝑥) 𝑄(𝑥) (𝑥 ∗ 𝑦) = log 𝑎 𝑥 + log 𝑎 𝑦 1 𝑛 ∫ 𝑏 𝑛 𝑎 √𝑘 + 𝑏 √𝑘 + 𝑐 √𝑘 = (𝑎 + 𝑏 + 𝑐) √𝑘 𝑛 𝑥 𝑎 𝑚 √√𝑎= 𝑛−𝑚 √𝑎 𝑅(𝑥) 𝑄(𝑥) 𝑑𝑥
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