Question Number 180873 by Shrinava last updated on 18/Nov/22 $$\mathrm{If}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c}<\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{abc}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)=\mathrm{64} \\ $$$$\mathrm{Then}\:\mathrm{find}\:\mathrm{min}\:\mathrm{of}\:\:\:\mathrm{P}=\mathrm{2a}+\mathrm{b}+\mathrm{c} \\ $$ Commented by mr W last updated on 18/Nov/22 $${please}\:{check}\:{the}\:{question}.\:{since} \\ $$$${a},{b},{c}<\mathrm{0},\:{P}\:{has}\:{maximum},\:{not}\:{min}.…
Question Number 115333 by zakirullah last updated on 25/Sep/20 Commented by zakirullah last updated on 25/Sep/20 $${solve}\:{only}\:{Q}\left(\mathrm{5},\mathrm{6}\right) \\ $$ Commented by bemath last updated on…
Question Number 115325 by zakirullah last updated on 25/Sep/20 Commented by mohammad17 last updated on 25/Sep/20 $$\left.{Q}\mathrm{5}/{i}\right) \\ $$$$ \\ $$$${z}^{\mathrm{2}} +{z}+\mathrm{3}=\mathrm{0}\Rightarrow{z}=\frac{−{b}\pm\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}{\mathrm{2}{a}}\Rightarrow{z}=\frac{−\mathrm{1}\pm{i}\sqrt{\mathrm{11}}}{\mathrm{2}} \\ $$$$\therefore{z}=−\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{11}}}{\mathrm{2}}{i}\:\:,\:{z}=−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{11}}}{\mathrm{2}}{i}…
Question Number 180856 by Shrinava last updated on 18/Nov/22 $$\mathrm{find}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{m}\in\mathbb{R}\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\boldsymbol{\mathrm{x}}} \:\frac{\mathrm{arctan}\boldsymbol{\mathrm{y}}}{\mathrm{y}}\:\mathrm{dy}\:=\:\mathrm{mx} \\ $$$$\mathrm{has}\:\mathrm{two}\:\mathrm{real}\:\mathrm{roots}:\:\:\:\mathrm{x}_{\mathrm{1}} \in\left(−\infty;\mathrm{0}\right)\:,\:\mathrm{x}_{\mathrm{2}} \in\left(\mathrm{0};\infty\right) \\ $$ Answered by mr W last…
Question Number 49755 by ajfour last updated on 10/Dec/18 $${Solve}\:{simultaneously}\:{for}\:\boldsymbol{{s}}\:{in}\:{terms} \\ $$$${of}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}. \\ $$$${h}^{\mathrm{2}} +\left({b}−{k}\right)^{\mathrm{2}} =\:{s}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:…..\left({i}\right) \\ $$$$\frac{{h}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{k}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..\left({ii}\right) \\ $$$$\left({h}−\frac{{s}}{\mathrm{2}}\right)^{\mathrm{2}}…
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Question Number 180785 by Vynho last updated on 17/Nov/22 $${solve}\:\left(\mathrm{1}−{x}−{x}^{\mathrm{2}} …\right)\left(\mathrm{2}−{x}−{x}^{\mathrm{2}} …\right) \\ $$ Answered by Rasheed.Sindhi last updated on 17/Nov/22 $$\left(\:\mathrm{1}−\left({x}+{x}^{\mathrm{2}} +…\right)\:\right)\left(\:\mathrm{2}−\left({x}+{x}^{\mathrm{2}} +…\right)\:\right) \\…
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Question Number 180759 by depressiveshrek last updated on 16/Nov/22 $${Find}\:{all}\:{functions}\:{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:{for}\:{all}\:{x},\:{y}\:\in\mathbb{R}\:{such}\:{that} \\ $$$${f}\left({f}\left({x}−{y}\right)−{yf}\left({x}\right)\right)={xf}\left({y}\right) \\ $$ Answered by aleks041103 last updated on 17/Nov/22 $${x}=\mathrm{0} \\ $$$$\Rightarrow{f}\left({f}\left(−{y}\right)−{yf}\left(\mathrm{0}\right)\right)=\mathrm{0} \\…
Question Number 115218 by Ar Brandon last updated on 24/Sep/20 $$\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N},\:\forall\mathrm{u}_{\mathrm{0}} ,\mathrm{u}_{\mathrm{1}} ,…,\mathrm{u}_{\mathrm{n}} ,\mathrm{v}_{\mathrm{0}} ,\mathrm{v}_{\mathrm{1}} ,…\mathrm{v}_{\mathrm{n}} \in\mathbb{C} \\ $$$$\forall\mathrm{k}\leqslant\mathrm{n};\:\mathrm{u}_{\mathrm{k}} =\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{k}} \begin{pmatrix}{\mathrm{k}}\\{\mathrm{i}}\end{pmatrix}\mathrm{v}_{\mathrm{i}} \Leftrightarrow\forall\mathrm{k}\leqslant\mathrm{n};\:\mathrm{v}_{\mathrm{k}} =\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{k}}…