Question Number 3451 by prakash jain last updated on 13/Dec/15 $$\mathrm{Consider}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{equation} \\ $$$$\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{i}} {x}^{{i}} =\mathrm{0},\:{a}_{{i}} \in\mathbb{Z} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{a}+{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{equation}\:\mathrm{then}\:{a}−{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{root}. \\ $$$${a},{b},{c}\in\mathbb{Z},\:{c}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{whole}\:\mathrm{square}. \\…
Question Number 68960 by ajfour last updated on 17/Sep/19 Commented by TawaTawa last updated on 17/Sep/19 $$\mathrm{Wow},\:\mathrm{weldone}\:\mathrm{sir},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$ Commented by ajfour last updated on…
Question Number 68935 by Maclaurin Stickker last updated on 17/Sep/19 $${Find}\:{all}\:{values}\:{for}\:\boldsymbol{{x}}: \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{11}\right)^{{x}^{\mathrm{2}} −\mathrm{13}{x}+\mathrm{42}} =\mathrm{1} \\ $$$$\left({Easy}\right) \\ $$ Answered by $@ty@m123 last updated…
Question Number 134467 by bramlexs22 last updated on 04/Mar/21 $$\begin{array}{|c|c|}{\mathrm{If}\:\mathrm{x}^{\mathrm{2}} =\mathrm{2y}+\mathrm{5}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} =\mathrm{2x}+\mathrm{5}\:}\\{\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} −\mathrm{2x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} .}\\\hline\end{array} \\ $$ Answered by benjo_mathlover last updated on 04/Mar/21…
Question Number 68912 by TawaTawa last updated on 16/Sep/19 Answered by MJS last updated on 17/Sep/19 $$\left(\mathrm{1}\right)\:\:{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} −\mathrm{7}=\mathrm{0} \\ $$$$\left(\mathrm{2}\right)\:\:{b}^{\mathrm{2}} +{bc}+{c}^{\mathrm{2}} −\mathrm{21}=\mathrm{0} \\ $$$$\left(\mathrm{3}\right)\:\:{c}^{\mathrm{2}}…
Question Number 68898 by Rio Michael last updated on 16/Sep/19 $${solve}\:{for}\:{x}\:{the}\:{equation} \\ $$$$\:\:{log}_{{x}} {e}^{\mathrm{2}{x}} \:=\:{eln}\:{x}\:−{e} \\ $$ Commented by kaivan.ahmadi last updated on 16/Sep/19 $$\frac{\mathrm{2}{x}}{{lnx}}={elnx}−{e}\Rightarrow{e}\left({lnx}\right)^{\mathrm{2}}…
Question Number 68899 by Rio Michael last updated on 16/Sep/19 $${solve}\:{for}\:{x}\:{and}\:{y}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{lnx}\:−{lny}\:={ln}\left(\mathrm{5}{x}−\mathrm{6}{y}\right) \\ $$ Commented by kaivan.ahmadi last updated on 16/Sep/19 $${i}\:{dont}\:{know}.{i}\:{think}\:{it}\:{is}\:{hard}. \\ $$…
Question Number 134410 by EDWIN88 last updated on 03/Mar/21 $$\:\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{xy}=\mathrm{2}}\\{{x}^{\mathrm{2}} +{z}^{\mathrm{2}} +{zx}=\mathrm{1}}\\{{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{yz}=\mathrm{1}}\end{cases}\:\Rightarrow\:\left({x}+{y}+{z}\right)^{\mathrm{2}} \:=? \\ $$ Commented by EDWIN88 last updated on…
Question Number 68835 by MJS last updated on 16/Sep/19 $$\mathrm{reposting}\:\mathrm{this}\:\mathrm{because}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{yet}\:\mathrm{solved} \\ $$$$\mathrm{correctly}.\:\mathrm{I}\:\mathrm{criticize}\:\mathrm{that}\:\mathrm{most}\:\mathrm{of}\:\mathrm{you}\:\mathrm{people} \\ $$$$\mathrm{do}\:\mathrm{not}\:\mathrm{test}\:\mathrm{if}\:\mathrm{your}\:\mathrm{solutions}\:\mathrm{fit}\:\mathrm{the}\:\mathrm{given} \\ $$$$\mathrm{equations}\:\mathrm{in}\:\mathrm{many}\:\mathrm{cases} \\ $$$$ \\ $$$${x}^{\mathrm{2}} +\mathrm{1}−\sqrt{{x}^{\mathrm{3}} +{x}}=\mathrm{6}{x} \\ $$$$ \\…
Question Number 134335 by bramlexs22 last updated on 02/Mar/21 $$ \\ $$Two Team A&B of workers at a bridge site. Each team consist of W…