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Category: Algebra

Solve-for-real-numbers-x-2-3n-2-4n-2-4-2-n-y-1-x-2-3n-2-4n-2-4-2-n-z-1-x-2-3n-2-4n-2-4-2-n-x-1-n-0-fixed-

Question Number 160712 by HongKing last updated on 05/Dec/21 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\left(\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{3n}^{\mathrm{2}} }{\mathrm{4n}^{\mathrm{2}} }\right)^{\mathrm{4}} =\:\frac{\mathrm{2}}{\mathrm{n}}\:\mathrm{y}-\mathrm{1}\:\:;\:\:\left(\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{3n}^{\mathrm{2}} }{\mathrm{4n}^{\mathrm{2}} }\right)^{\mathrm{4}} =\:\frac{\mathrm{2}}{\mathrm{n}}\:\mathrm{z}-\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{3n}^{\mathrm{2}} }{\mathrm{4n}^{\mathrm{2}} }\right)^{\mathrm{4}}…

Be-p-a-prime-number-arbitrary-Solve-on-positive-integers-x-y-z-xy-z-2-3p-4-x-yz-2-p-4-

Question Number 160711 by HongKing last updated on 05/Dec/21 $$\mathrm{Be}\:\:\boldsymbol{\mathrm{p}}\:\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:,\:\mathrm{arbitrary}. \\ $$$$\mathrm{Solve}\:\mathrm{on}\:\mathrm{positive}\:\mathrm{integers}\:\:\left(\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}};\boldsymbol{\mathrm{z}}\right) \\ $$$$\begin{cases}{\mathrm{xy}\:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{3p}\:+\:\mathrm{4}}\\{\mathrm{x}\:+\:\mathrm{yz}^{\mathrm{2}} \:=\:\mathrm{p}\:+\:\mathrm{4}}\end{cases} \\ $$ Answered by mr W last updated on…

the-first-term-in-a-geometric-series-is-2x-7-2x-5-and-the-common-ratio-is-2x-5-2x-7-find-the-set-of-values-of-x-for-which-all-the-terms-are-possible-

Question Number 95167 by Rio Michael last updated on 23/May/20 $$\mathrm{the}\:\mathrm{first}\:\mathrm{term}\:\mathrm{in}\:\mathrm{a}\:\mathrm{geometric}\:\mathrm{series}\:\mathrm{is}\:\frac{\left(\mathrm{2}{x}\:+\:\mathrm{7}\right)}{\mathrm{2}{x}−\mathrm{5}}\:\mathrm{and}\:\mathrm{the}\:\mathrm{common}\:\mathrm{ratio}\:\mathrm{is} \\ $$$$\:\frac{\left(\mathrm{2}{x}−\mathrm{5}\right)}{\mathrm{2}{x}\:+\:\mathrm{7}}\:\mathrm{find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{for}\:\mathrm{which}\:\mathrm{all}\:\mathrm{the}\:\mathrm{terms}\:\mathrm{are}\:\mathrm{possible}. \\ $$ Commented by john santu last updated on 23/May/20 $$\left(\mathrm{1}\right)\:\mathrm{x}\:\neq\:\frac{\mathrm{5}}{\mathrm{2}}\:\wedge\:\mathrm{x}\:\neq\:−\frac{\mathrm{7}}{\mathrm{2}}\: \\…

8-x-1-3-x-2-

Question Number 95159 by i jagooll last updated on 23/May/20 $$\sqrt[{\mathrm{3}\:\:}]{\mathrm{8}−\mathrm{x}}\:+\:\sqrt{\mathrm{x}}\:=\:\mathrm{2}\: \\ $$ Commented by hknkrc46 last updated on 23/May/20 $$\bigstar\:\sqrt[{\mathrm{2}{n}+\mathrm{1}}]{{f}\left({x}\right)}\:\Rightarrow\:\forall{f}\left({x}\right)\:\in\:\mathbb{R}\:\rightarrow\:{f}\left({x}\right)\geqslant\mathrm{0}\:\vee\:{f}\left({x}\right)\leqslant\mathrm{0} \\ $$$$\bigstar\sqrt[{\mathrm{2}{n}}]{{f}\left({x}\right)}\:\Rightarrow\:\forall{f}\left({x}\right)\:\in\:\mathbb{R}^{+} \:\rightarrow\:{f}\left({x}\right)\geqslant\mathrm{0} \\…