Question Number 211605 by MrGaster last updated on 14/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Let}\:\boldsymbol{{a}}_{\mathrm{1}} ,\boldsymbol{{a}}_{\mathrm{2}} ,\ldots\boldsymbol{{a}}_{\boldsymbol{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Is}\:\mathrm{n}\:\mathrm{real}\:\mathrm{numbers}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{All}\:\mathrm{fall}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left(−\mathrm{1},\mathrm{1}\right) \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\left(\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{1}\leq\boldsymbol{{i}},\boldsymbol{{j}}\leq\boldsymbol{{n}}} {\prod}\frac{\mathrm{1}+\boldsymbol{{a}}_{\boldsymbol{{i}}} \boldsymbol{{a}}_{\boldsymbol{{j}}} }{\mathrm{1}−\boldsymbol{{a}}_{\boldsymbol{{i}}}…
Question Number 211601 by MrGaster last updated on 14/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\mathrm{set}\:\boldsymbol{\alpha}\sqrt[{\mathrm{3}}]{\mathrm{2}},\mathrm{ask}\:\mathbb{Q}\left(\boldsymbol{\alpha}\right)\mathrm{Upper}\:\mathrm{irreducible}\:\mathrm{cubic}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{x}}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{All}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 211617 by pete last updated on 14/Sep/24 $$\mathrm{Ato}\:\:\mathrm{starts}\:\mathrm{a}\:\mathrm{business}\:\mathrm{with}\:\$\mathrm{1},\mathrm{250}.\mathrm{00}.\:\mathrm{Ama}\:\mathrm{joins}\:\mathrm{the}\:\mathrm{business} \\ $$$$\mathrm{later}\:\mathrm{with}\:\mathrm{a}\:\mathrm{capital}\:\mathrm{of}\:\mathrm{1},\mathrm{875}.\mathrm{00}.\:\mathrm{At}\:\mathrm{the}\: \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{year},\:\mathrm{profits}\:\mathrm{are}\:\mathrm{shared}\:\mathrm{equally} \\ $$$$\mathrm{between}\:\mathrm{Ato}\:\mathrm{and}\:\mathrm{Ama}.\:\mathrm{When}\:\mathrm{did}\:\mathrm{Ama} \\ $$$$\mathrm{join}\:\mathrm{the}\:\mathrm{business}? \\ $$ Answered by A5T last updated…
Question Number 211574 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{25}}\\{\boldsymbol{{x}}+\mathrm{2}\boldsymbol{\mathrm{y}}−\mathrm{3}=\mathrm{0}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$ Answered by Rasheed.Sindhi…
Question Number 211548 by MrGaster last updated on 12/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{y}}^{\mathrm{2}} −\mathrm{6}\boldsymbol{{x}\mathrm{y}}=\mathrm{12}}\\{\boldsymbol{{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{4}}\end{cases} \\ $$$$ \\ $$ Answered by Frix last updated…
Question Number 211535 by hardmath last updated on 12/Sep/24 Answered by BaliramKumar last updated on 12/Sep/24 $${x}\:=\:\mathrm{7}\:−\:\frac{\mathrm{6}}{{x}} \\ $$$${x}\:=\:\left(\mathrm{1},\:\mathrm{6}\right) \\ $$ Terms of Service Privacy…
Question Number 211524 by Tawa11 last updated on 12/Sep/24 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}:\:\:\:\mathrm{x}^{\mathrm{x}} \:\:=\:\:\mathrm{4x} \\ $$ Answered by Frix last updated on 12/Sep/24 $$\mathrm{We}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate}. \\ $$$${x}\approx.\mathrm{183193} \\ $$$${x}\approx\mathrm{2}.\mathrm{50781}…
Question Number 211509 by RojaTaniya last updated on 11/Sep/24 Answered by A5T last updated on 11/Sep/24 $$\frac{\mathrm{6}{a}+{b}}{\mathrm{6}{c}+{d}}={k}=\frac{\mathrm{5}{a}+{b}}{\mathrm{5}{c}+{d}} \\ $$$$\Rightarrow\mathrm{6}{a}+{b}=\mathrm{6}{ck}+{dk}…\left({i}\right);\mathrm{5}{a}+{b}=\mathrm{5}{ck}+{dk}…\left({ii}\right) \\ $$$$\left({i}\right)−\left({ii}\right)\Rightarrow{a}={ck};\mathrm{6}\left({ii}\right)−\mathrm{5}\left({i}\right)\Rightarrow{b}={dk} \\ $$$$\mathrm{7}{a}+{b}=\mathrm{8}\left(\mathrm{7}{c}+{d}\right)\Rightarrow\mathrm{7}{ck}+{dk}=\mathrm{8}\left(\mathrm{7}{c}+{d}\right) \\ $$$$\Rightarrow{k}=\frac{\mathrm{8}\left(\mathrm{7}{c}+{d}\right)}{\mathrm{7}{c}+{d}}=\mathrm{8}…
Question Number 211520 by a.lgnaoui last updated on 12/Sep/24 $$\mathrm{determiner}\:\mathrm{les}\:\mathrm{valeurs}\:\:\mathrm{de}\:\boldsymbol{\mathrm{p}}\mathrm{et}\:\boldsymbol{\mathrm{q}}\:\mathrm{sachant}\:\mathrm{que}\:−\mathrm{2}\:\mathrm{et}\:\mathrm{3}\:\mathrm{sont}\:\mathrm{les}\: \\ $$$$−\mathrm{2}\:\mathrm{et}\:\mathrm{3}\:\:\mathrm{sont}\:\mathrm{les}\:\mathrm{racines}\:\mathrm{de}\:\mathrm{l}\:\mathrm{equation}: \\ $$$$\mathrm{2}\boldsymbol{\mathrm{pqz}}^{\mathrm{2}} −\mathrm{5}\boldsymbol{\mathrm{z}}−\mathrm{4}\left(\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}}\right)=\mathrm{0} \\ $$$$ \\ $$ Commented by MrGaster last updated on…
Question Number 211485 by MATHEMATICSAM last updated on 10/Sep/24 $$\mathrm{If}\:\frac{{a}\:−\:{b}}{{c}}\:+\:\frac{{b}\:−\:{c}}{{a}}\:+\:\frac{{c}\:+\:{a}}{{b}}\:=\:\mathrm{1}\:\mathrm{and}\: \\ $$$${a}\:−\:{b}\:+\:{c}\:\neq\:\mathrm{0}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}}\:=\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:. \\ $$ Answered by Frix last updated on 10/Sep/24 $$\frac{{a}−{b}}{{c}}+\frac{{b}−{c}}{{a}}+\frac{{c}+{a}}{{b}}=\mathrm{1} \\…