Question Number 203716 by professorleiciano last updated on 26/Jan/24 Answered by AST last updated on 26/Jan/24 $${Let}\:{AP}\:{and}\:{DC}\:{meet}\:{at}\:{F},{then}\:\bigtriangleup{BPA}\approx\bigtriangleup{CPF} \\ $$$$\Rightarrow\frac{{BA}}{{CF}}=\frac{{BP}}{{CP}}\Rightarrow\frac{{s}}{{CF}}=\frac{\mathrm{8}}{{s}−\mathrm{8}}\Rightarrow{CF}=\frac{{s}\left({s}−\mathrm{8}\right)}{\mathrm{8}}…\left({i}\right) \\ $$$${in}\:\bigtriangleup{AFD};{AQ}\:{bisects}\:\angle{FAD}\Rightarrow\frac{{AD}}{{DQ}}=\frac{{AF}}{{FQ}} \\ $$$$\Rightarrow\frac{{s}}{\mathrm{9}}=\frac{\sqrt{{s}^{\mathrm{2}} +\left({s}+{CF}\right)^{\mathrm{2}} }}{{CF}+{s}−\mathrm{9}}\Rightarrow\frac{{s}}{\mathrm{9}}=\frac{\sqrt{{s}^{\mathrm{2}}…
Question Number 203712 by Davidtim last updated on 26/Jan/24 $$\mathrm{50}\:{divide}\:{into}\:{to}\:{part}\:{that}\:{the}\:{mini}\: \\ $$$${figure}\:\frac{\mathrm{2}}{\mathrm{3}}\:{and}\:{large}\:{one}\:{is}\:\frac{\mathrm{5}}{\mathrm{7}},\:{and}\:{the}\: \\ $$$${sum}\:{of}\:{them}\:{is}\:\mathrm{21},\:{find}\:{the}\:{tow}\: \\ $$$${figures}? \\ $$ Commented by Davidtim last updated on 30/Jan/24…
Question Number 203714 by K1000 last updated on 26/Jan/24 $$\int\mathrm{2}{x}^{\mathrm{2}} \\ $$ Answered by Frix last updated on 26/Jan/24 $$\int{ax}^{{n}} {dx}={a}\int{x}^{{n}} {dx}=\frac{{ax}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}+{C} \\ $$…
Question Number 203676 by 073 last updated on 25/Jan/24 Answered by Frix last updated on 25/Jan/24 $${x}\left({x}+\mathrm{2}\right){x}!=\mathrm{168}{x}^{\mathrm{2}} \\ $$$${x}=\mathrm{0} \\ $$$$\left({x}+\mathrm{2}\right){x}!=\mathrm{168}{x} \\ $$$$\left({x}+\mathrm{2}\right)\left({x}−\mathrm{1}\right)!=\mathrm{168}=\mathrm{2}×\mathrm{3}×\mathrm{4}×\mathrm{7} \\ $$$${x}=\mathrm{5}…
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Question Number 203675 by 073 last updated on 25/Jan/24 Answered by Frix last updated on 25/Jan/24 $${x}^{\mathrm{4}} −\mathrm{2}{ax}^{\mathrm{2}} −{x}+{a}^{\mathrm{2}} −{a}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} +{x}−{a}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}−{a}\right)=\mathrm{0} \\…
Question Number 203668 by Perelman last updated on 25/Jan/24 Answered by witcher3 last updated on 25/Jan/24 $$\mathrm{x}\overset{\mathrm{f}} {\rightarrow}\frac{\mathrm{1}}{\mathrm{4}−\mathrm{3x}} \\ $$$$\mathrm{f}'\left(\mathrm{x}\right)=\frac{\mathrm{3}}{\left(\mathrm{4}−\mathrm{3x}\right)^{\mathrm{2}} } \\ $$$$\forall\mathrm{x}\in\left[−\mathrm{1},\frac{\mathrm{1}}{\mathrm{3}}\right]\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\in\left[\frac{\mathrm{1}}{\mathrm{7}},\frac{\mathrm{1}}{\mathrm{3}}\right]…..\left(\mathrm{1}\right) \\ $$$$\left.\Rightarrow\mathrm{f}\left[−\mathrm{1},\frac{\mathrm{1}}{\mathrm{3}}\right]\right)\subset\left[\frac{\mathrm{1}}{\mathrm{7}},\frac{\mathrm{1}}{\mathrm{3}}\right]…
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