Question Number 11515 by @ANTARES_VY last updated on 27/Mar/17 $$\sqrt[{\mathrm{5}}]{\frac{\mathrm{4}+\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}}−\sqrt[{\mathrm{5}}]{\frac{\mathrm{4}−\mathrm{6}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}}=\mathrm{1}. \\ $$$$\boldsymbol{\mathrm{x}}_{\mathrm{1}} +\boldsymbol{\mathrm{x}}_{\mathrm{2}} =? \\ $$ Answered by ajfour last updated on 27/Mar/17 $$\mathrm{x}=\frac{\mathrm{4}}{\mathrm{15}} \\…
Question Number 142577 by loveineq last updated on 02/Jun/21 $$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{1}.\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{1}}{{b}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{1}}{{c}^{\mathrm{2}} +\mathrm{1}}\:\geqslant\:\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$ \\ $$ Answered by ajfour last updated on…
Question Number 142582 by mathdanisur last updated on 02/Jun/21 $$\boldsymbol{{z}}^{\mathrm{4}} \:+\:\mathrm{12}\boldsymbol{{z}}\:+\:\mathrm{3}\:=\:\mathrm{0}\:\left(\boldsymbol{{z}}=?\right) \\ $$ Answered by ajfour last updated on 02/Jun/21 $$\left({z}^{\mathrm{2}} +{pz}+{q}\right)\left({z}^{\mathrm{2}} −{pz}+\frac{\mathrm{3}}{{q}}\right)=\mathrm{0} \\ $$$$\Rightarrow…
Question Number 11500 by @ANTARES_VY last updated on 27/Mar/17 $$\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}−\mathrm{4}\right)\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}+\mathrm{4}\right)=\mathrm{9} \\ $$$$\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{equation}}\:\:\boldsymbol{\mathrm{has}}\:\:\boldsymbol{\mathrm{multiple}}\:\:\boldsymbol{\mathrm{roots}}. \\ $$ Answered by ridwan balatif last updated on 27/Mar/17 $$\left(\left(\mathrm{x}^{\mathrm{2}}…
Question Number 11501 by @ANTARES_VY last updated on 27/Mar/17 $$\left(\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{7}\boldsymbol{\mathrm{x}}−\mathrm{5}\right)\left(\mathrm{5}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{13}\boldsymbol{\mathrm{x}}+\mathrm{3}\right)\left(\mathrm{3}\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{8}\right)=\mathrm{0}. \\ $$$$\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{all}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{multiples}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{real}}\:\:\boldsymbol{\mathrm{roots}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{equation}}. \\ $$ Commented by mrW1 last updated on…
Question Number 142570 by mathsuji last updated on 02/Jun/21 $${Find}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{such}\:{that} \\ $$$${f}\left({x}+{y}\right)=\mathrm{2}{f}\left({x}\right)+\mathrm{3}{f}\left({y}\right)−\mathrm{4}{xyf}\left(\mathrm{2}{x}−\mathrm{3}{y}\right) \\ $$$$\left(\forall{x};{y}\in\mathbb{R}\right) \\ $$ Answered by ajfour last updated on 02/Jun/21 $${f}\left({x}\right)=\mathrm{5}{f}\left(\frac{{x}}{\mathrm{2}}\right)−{x}^{\mathrm{2}} {f}\left(−\frac{{x}}{\mathrm{2}}\right)…
Question Number 77035 by necxxx last updated on 02/Jan/20 $${Solve}\:{for}\:{x}\:{in}: \\ $$$$\left({i}\right)\:\left(\mathrm{2}\left({x}+\mathrm{3}\right)−\mathrm{3}\left({x}−\mathrm{2}\right)\right)\left(\mathrm{2}{x}−\mathrm{1}\right)\geqslant\mathrm{0} \\ $$$$\left({ii}\right)\left({x}−\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{3}\right)\left({x}+\mathrm{1}\right)\left({x}+\mathrm{3}\right)\leqslant\mathrm{1} \\ $$ Answered by MJS last updated on 03/Jan/20 $$\left({i}\right) \\…
Question Number 77028 by TawaTawa last updated on 02/Jan/20 Answered by mind is power last updated on 02/Jan/20 $$\mathrm{39} \\ $$$$\mathrm{x}−\mathrm{y}\:−\mathrm{z}\left(\mathrm{x}−\mathrm{y}\right)=\mathrm{0} \\ $$$$\mathrm{y}−\mathrm{z}−\mathrm{x}\left(\mathrm{y}−\mathrm{z}\right)=\mathrm{0} \\ $$$$\mathrm{x}^{\mathrm{2}}…
Question Number 77033 by necxxx last updated on 02/Jan/20 $${Suppose}\:{the}\:{population}\:{models}\:{of}\:{London} \\ $$$${and}\:{Hongkong}\:{in}\:{tens}\:{of}\:{thousands}\:{are} \\ $$$${p}\left({t}\right)=\frac{\mathrm{20}{t}}{{t}+\mathrm{1}}\:{and}\:{q}\left({t}\right)=\frac{\mathrm{240}{t}}{{t}+\mathrm{8}}\:{respectively}\:{for} \\ $$$${t}\:{years}\:{after}\:\mathrm{2015},\:{Determine}\:{the}\:{time}\:{period} \\ $$$${in}\:{years}\:{when}\:{the}\:{population}\:{of}\:{London} \\ $$$${exceeds}\:{that}\:{of}\:{HongKong}. \\ $$ Commented by MJS…
Question Number 142556 by iloveisrael last updated on 02/Jun/21 $$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}{yz}+\mathrm{2}−{z}^{\mathrm{2}} }\\{{z}=\mathrm{4022}−{x}−{y}}\end{cases} \\ $$$$\:{x},{y},{z}\in\mathbb{Z}^{+} \:.\:{then}\:{z}= \\ $$ Answered by iloveisrael last updated on 02/Jun/21…