Category: Algebra

Question Number 227016 by Spillover last updated on 25/Dec/25 $${If}\:\alpha\:{and}\:.\beta\:\:{are}\:{root}\:{of}\:{the}\:{equation} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{4}=\mathrm{0}\:{and}\:{for}\:{n}\in\mathbb{N} \\ $$$${Show}\:{that}\: \\ $$$$\alpha^{{n}} +\beta^{{n}} =\mathrm{2}^{{n}+\mathrm{1}} \mathrm{cos}\:\left(\frac{{n}\pi}{\mathrm{3}}\right) \\ $$ Answered by Frix…

Question Number 226983 by hardmath last updated on 24/Dec/25 Answered by MrAjder last updated on 01/Jan/26 $$ \\ $$$${S}=\left\{\left({a},{b},{c}\right)\in\mathbb{R}_{+} ^{\mathrm{3}} :{a}+{b}+{c}=\mathrm{3}\right\},{F}\left({a},{b},{c}\right)=\underset{{cyc}} {\sum}\frac{{a}}{{b}^{\mathrm{2}} +{c}} \\ $$$$\mathcal{L}\left({a},{b},{c},\lambda\right)={F}\left({a},{b},{c}\right)+\lambda\left({a}+{b}+{c}−\mathrm{3}\right)…

Question Number 226958 by efronzo1 last updated on 21/Dec/25 $$\:\:\:\mathrm{3}^{\mathrm{x}} =\mathrm{x}^{\mathrm{9}} \: \\ $$$$\:\:\:\:\mathrm{x}^{\mathrm{2}} =\:..? \\ $$ Answered by TonyCWX last updated on 21/Dec/25 $${x}\mathrm{ln}\left(\mathrm{3}\right)\:=\:\mathrm{9ln}\left({x}\right)…

Question Number 226912 by Spillover last updated on 18/Dec/25 Answered by Raphael254 last updated on 19/Dec/25 $$ \\ $$$${b}+{c}\:=\:\mathrm{16} \\ $$$$ \\ $$$${a}+{b}+{c}=\mathrm{24} \\ $$$${a}+\mathrm{16}\:=\:\mathrm{24}…

Question Number 226879 by Spillover last updated on 17/Dec/25 Answered by gregori last updated on 18/Dec/25 $$\left({a}\right)\:{f}\:'\left({x}\right)=\:−\mathrm{sin}\:{x}\:.\:{e}^{\mathrm{cos}\:{x}} \: \\ $$$$\:\left({b}\right)\:{f}\:'\left({x}\right)\:=\:\mathrm{3sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:{x}\:.{e}^{\mathrm{sin}\:^{\mathrm{3}} {x}} \\ $$$$ \\…