Pavel Durov one

1️⃣3️⃣ Telegram turns 13 today. For over half that time — the past 7,5 years — it's been one of the 10 most downloaded mobile apps in the world. A whole generation has grown up with it 💪 🎂 Happy birthday, Telegram!

❄️Notes on Quadratic Equations⛄️ 📱In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac. 🎁The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a 🎁If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots: α + β = -b/a α.β = c/a α – β = √D/a 🎁It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity. Nature of Roots: 🎁Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases: D > 0 iff the roots are real and distinct i.e. the roots are unequal D = 0 iff the roots are real and coincident i.e. equal D < 0 iffthe roots are imaginary The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1. 🎁Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then If D > 0 and is also a perfect square then the roots are rational and unequal. 🎁If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa. If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula x2 – (Sum of roots)x + (Product of roots) = 0. So if α and β are the roots of equation then the quadratic equation is x2 – (α + β)x + α β = 0 For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola. If a > 0, then the shape of the parabola is concave upwards If a < 0, then the shape of the parabola is concave upwards Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule. The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0. y ∈[(4ac-b2) / 4a, ∞] if a > 0 If a < 0, then y ∈ [-∞, (4ac-b2) / 4a] The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0 In general, if α1,α2, α3, …… ,αn are the roots of the equation f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then 1.Σα1 = - a1/a0 2.Σ α1α2 = a2/a0 3.Σ α1α2α3 = - a3/a0 ………   ……….   Σ α1α2α3 ……αn= (-1)n an/a0 Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity. If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’. 🎁Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.

⏳ Kanal egasi D ? bir haftadan beri asosiy botga yozmadi — shu sababdan botlar reaksiya bosishni to'xtatdi. 👇 Asosiy botlardan biriga /start deb yozing: 🤖 @ReaksiyalarBot 🤖 @ReaksiyaBosBot ✔️ Yozganingizdan keyin reaksiyalar tiklanadi va bu xabar o'chadi.

🇺🇿 The 2026 International Olympiad in Informatics starts today in Tashkent. 🏆 The IOI is the ultimate coding competition for high-school students. Years ago, many members of the Telegram team won medals there. 🤝 To support the best young programmers in the world, Telegram will award the 235 winners exclusive digital collectibles — Algorithm Cups — worth at least $117,000 in total. 🕌 It’s fitting that IOI is hosted in Uzbekistan — the homeland of Al-Khwarizmi, the father of algebra whose name became the word “algorithm." Let the best algorithm win! 💪

╰┈➤ Teclado iOS 27 para Android 📆 Fecha: 09-07-2026 ⭐️ Base: 3.0.2 ✧ LootLink: ➣ •Descargar | •Download | •Baixar ✧ Cutly: ➣ •Descargar | •Download | •Baixar 💡 CÓMO DESCARGAR LOS MODS DEL CANAL? Aquí tienes un: 🎥 TUTORIAL ---------------------------- ➥ Características y tips del mod ➡️ Instalación,configuración y vista previa del mod. 📸 SCREENSHOTS 😵 ¿Cansado(a) de los anuncios al descargar mods? Únete al VIP de iOSStarWorld y recibe: ✅ Links directos sin publicidad ✅ Mods anticipados ✅ Apps,temas premium y  mucho más! 📩 Escríbele a @TheAngelM23 para obtener más información. ---------------------------------- ✅ ÚNETE AL iOS STARWORLD TEAM  ------------------------- ✅ Para informes de errores o solicitudes de aplicaciones, únanse a nuestra comunidad: •https://t.me/iosstarworldcommunity_official ✰ Base de Best iOS Setups & Modificado por el iOS StarWorld Team ✰ ➡️ Reacciona a los posts del canal para seguir subiendo más contenido!!! #iOS #KeyBoard #TecladoiOS #iOS26 #iOSStarWorld

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🔐 We’ve upgraded Telegram’s anti-censorship protocol. Users in Russia are advised to update their apps to stay connected despite the ban. ⚠️ Some other advice: stock up on several VPNs/proxies in advance, help your friends and family do the same, and avoid using Russian apps while connected to a VPN (they may report your VPN to authorities for blocking). 😎  I’m glad to see most people are already doing this. Thanks to this Digital Resistance, Telegram usage in Russia has remained stable over the past week despite the full ban. 🔥 Keep it up. On our side, we’ll continue improving Telegram’s decentralized anti-censorship tech.

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🏆 The International Olympiad in AI finished yesterday in Astana. 🇷🇺 The Russian team took first place in the team competition, and Artem Gorokhov from my home city won the individual event. Well done! 💪 🇰🇿 I'm equally happy for the host country, Kazakhstan, whose Dauzhan Beketov took the second-highest individual score.👏 📩We're in touch with the organizers to deliver the winners' prizes from Telegram ($106,500+) securely next week. Congratulations to all the winners! 🏆

▶️ The auction is LIVE! 🥊 t.me/auction/ufcstrike

🎁 Our new Login Widget is a gift to developers: they can not only quickly authorize users, but also request users’ phone numbers and permission to message them on Telegram. 🆓 Sign-up, login, phone verification, and a follow-up communication channel in one — completely free.

#Advertising #Publicidad #iOSStarWorld ⚠️ 🇪🇸 El siguiente post es una PUBLICIDAD PAGADA ⭐️ ✉️ Si te interesa publicar algún contenido en especial para hacer conocer tu marca puedes escribir a @TheAngelM23 para solicitar información. ------------------------------------ ⚠️ 🇦🇽 The following post is a PAID ADVERTISING ⭐️ ✉️ If you are interested in publishing any special content to make your brand known, you can write to @TheAngelM23 to request information. ------------------------------------ ⚠️ 🇧🇷 O post a seguir é uma PUBLICIDADE PAGA 🔔 ✉️ Se você tiver interesse em publicar algum conteúdo especial para divulgar sua marca, pode escrever para @TheAngelM23 para solicitar informações. iOS StarWorld Team ⁩

╰┈➤ Telegram Estilo iOS 26 para Android 📆 Fecha: 23-07-2026 🏷 Base: 12.9.2 (69799) @durovme1 Edition ✧ LootLabs: ➣ •Descargar | •Download | •Baixar ✧ Cutly: ➣ •Descargar | •Download | •Baixar 💡 CÓMO DESCARGAR LOS MODS DEL CANAL? Aquí tienes un: 🎥 TUTORIAL ---------------------- 🔥 Instalación,configuración y vista previa del mod. 🌟 SCREENSHOTS ⭐️ ACTIVAR LIQUID GLASS 😵 ¿Cansado(a) de los anuncios al descargar mods? Únete al VIP de iOSStarWorld y recibe: ✅ Links directos sin publicidad ✅ Mods anticipados ✅ Apps,temas premium y mucho más! ✉️ Escríbele a @TheAngelM23 para obtener más información. ---------------------------- ✅ Para informes de errores o solicitudes de aplicaciones, únanse a nuestra comunidad: •https://t.me/iosstarworldcommunity_official ✰ Modificado por el iOS StarWorld Team ✰ ➡️ Reacciona a los posts del canal para seguir subiendo más contenido!!! #iOS #TelegramiOS #iOSStarWorld

╰┈➤ Apple Music Reprodutor de Música iOS 27 📆 Fecha: 30-07-2026 ⚙️ Base: 2.1.0 @iOSStarWorld Edition LootLink ✧ Flamingo: ➣ •Descargar | •Download | •Baixar ✧ MusicTag: ➣ •Descargar | •Download | •Baixar ✧ SongSync: ➣ •Descargar | •Download | •Baixar Cutly ✧ Flamingo: ➣ •Descargar | •Download | •Baixar ✧ MusicTag: ➣ •Descargar | •Download | •Baixar ✧ SongSync: ➣ •Descargar | •Download | •Baixar 💡 CÓMO DESCARGAR LOS MODS DEL CANAL? Aquí tienes un: 👾 TUTORIAL ----------------------------- ➡️ INSTALACIÓN Y CONFIGURACIÓN 😎 SCREENSHOTS 🆕 NUEVAS FUNCIONES 🎯 ¿Cansado(a) de los anuncios al descargar mods? Únete al VIP de iOSStarWorld y recibe: ✅ Links directos sin publicidad ✅ Apps,temas premium y  mucho más! ✉️ Escríbele a @TheAngelM23 para obtener más información. -------------------------------- 🍽 ÚNETE AL iOS STARWORLD TEAM  ----------------------- ✰ Modificado por el iOS StarWorld Team ✰ ➡️ Reacciona a los posts del canal para seguir subiendo más contenido!!! #iOS #Music #iOSStarWorld