Question Number 217262 by ArshadS last updated on 07/Mar/25 $${Solve}: \\ $$$$\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}}{{x}^{\mathrm{3}} −\mathrm{4}{x}}−\frac{\mathrm{2}}{{x}^{\mathrm{2}} +\mathrm{2}{x}}=\frac{\mathrm{1}}{{x}−\mathrm{2}} \\ $$ Answered by ibrahimmatematic last updated on 07/Mar/25 $$\frac{{x}\left({x}+\mathrm{3}\right)}{{x}\left({x}−\mathrm{2}\right)\left({x}+\mathrm{2}\right)}−\frac{\mathrm{2}}{{x}\left({x}+\mathrm{2}\right)}=\frac{\mathrm{1}}{{x}−\mathrm{2}}…
Question Number 217225 by hardmath last updated on 06/Mar/25 $$\mathrm{Find}:\:\:\:\:\:\int\:−\:\frac{{d}\mathrm{x}}{\:\sqrt{\mathrm{24x}\:−\:\mathrm{16x}^{\mathrm{2}} \:−\:\mathrm{8}}}\:=\:? \\ $$ Answered by Frix last updated on 06/Mar/25 $$−\int\frac{{dx}}{\:\sqrt{−\mathrm{16}{x}^{\mathrm{2}} +\mathrm{24}{x}−\mathrm{8}}}\:\overset{\left[{t}=\mathrm{sin}^{−\mathrm{1}} \:\left(\mathrm{4}{x}−\mathrm{3}\right)\right]} {=} \\…
Question Number 217197 by hardmath last updated on 05/Mar/25 $$\mathrm{Find}: \\ $$$$\sqrt[{\mathrm{5}}]{−\mathrm{32}}\:\:+\:\:\sqrt[{\mathrm{8}}]{\left(−\mathrm{3}\right)^{\mathrm{8}} }\:\:=\:\:? \\ $$ Answered by Rasheed.Sindhi last updated on 05/Mar/25 $$\sqrt[{\mathrm{5}}]{−\mathrm{32}}\:\:+\:\:\sqrt[{\mathrm{8}}]{\left(−\mathrm{3}\right)^{\mathrm{8}} }\: \\…
Question Number 217198 by hardmath last updated on 05/Mar/25 Answered by MrGaster last updated on 05/Mar/25 $$\delta\left({n}\right)=\underset{{d}\mid{n}} {\sum}{d},“\tau\left({n}\right)=\underset{{d}\mid{n}} {\sum}\cancel{{l}}\overset{\left(\mathrm{Unknown}\:\mathrm{meaning}\right)} {\Rightarrow}\mathrm{1}''\:\:\mathrm{and}\:\varphi-\mathrm{Euler}^{,} \mathrm{s}\:\mathrm{totient}\:\mathrm{function} \\ $$$$\delta\left({n}\right)=\underset{{i}=\mathrm{1}} {\overset{{k}} {\prod}}\left({p}_{{i}}…
Question Number 217205 by peter frank last updated on 05/Mar/25 Answered by som(math1967) last updated on 06/Mar/25 $$\mathrm{cos}\:\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{sin}\:\mathrm{cot}^{−\mathrm{1}} {x}\right) \\ $$$$=\mathrm{cos}\:\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{sin}\:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\right)…
Question Number 217203 by Rasheed.Sindhi last updated on 05/Mar/25 $$\mathrm{A}\:\mathrm{farmer}\:\mathrm{has}\:\mathrm{100}\:\mathrm{meters}\:\mathrm{of}\: \\ $$$$\mathrm{fencing}\:\mathrm{and}\:\mathrm{wants}\:\mathrm{to}\:\mathrm{enclose}\:\mathrm{an} \\ $$$$\mathrm{rectagular}\:\mathrm{field}\:\mathrm{along}\:\mathrm{a}\:\mathrm{river}.\:\mathrm{Thei} \\ $$$$\mathrm{rver}\:\mathrm{forms}\:\mathrm{one}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{rectangle}\:\mathrm{so}\:\mathrm{fencing}\:\mathrm{is}\:\mathrm{needed}\:\mathrm{onlyo} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{other}\:\mathrm{three}\:\mathrm{sides}.\:\mathrm{What}\: \\ $$$$\mathrm{dimesions}\:\mathrm{should}\:\mathrm{the}\:\mathrm{farmer}\: \\ $$$$\mathrm{chooseto}\:\mathrm{maximize}\:\mathrm{the}\:\mathrm{enclosed} \\…
Question Number 217178 by hardmath last updated on 04/Mar/25 $$\mathrm{Find}: \\ $$$$\mathrm{100}-\mathrm{99}+\mathrm{98}-\mathrm{97}+\mathrm{96}-\mathrm{95}+…+\mathrm{2}-\mathrm{1}\:=\:? \\ $$ Answered by MathematicalUser2357 last updated on 04/Mar/25 $$=\left(\mathrm{100}+\mathrm{98}+\mathrm{96}+…{choegangcheck}\:{is}\:{cute}…+\mathrm{6}+\mathrm{4}+\mathrm{2}\right)−\left(\mathrm{99}+\mathrm{97}+\mathrm{95}+…{sorry}\:{for}\:{aggro}…+\mathrm{5}+\mathrm{3}+\mathrm{1}\right) \\ $$$$=\mathrm{2}\left(\mathrm{50}+\mathrm{49}+\mathrm{48}+…+\mathrm{3}+\mathrm{2}+\mathrm{1}\right)−\left\{\mathrm{2}\left(\mathrm{50}+\mathrm{49}+\mathrm{48}+…+\mathrm{3}+\mathrm{2}+\mathrm{1}\right)−\mathrm{50}\right\} \\…
Question Number 217186 by MrGaster last updated on 04/Mar/25 $$\frac{{x}−\mathrm{4051}}{\mathrm{2024}}+\frac{{x}−\mathrm{4050}}{\mathrm{2025}}+\frac{{x}−\mathrm{4049}}{\mathrm{2026}}=\mathrm{3} \\ $$ Answered by som(math1967) last updated on 04/Mar/25 $$\:\frac{{x}−\mathrm{4051}}{\mathrm{2024}}\:−\mathrm{1}+\frac{{x}−\mathrm{4050}}{\mathrm{2025}}−\mathrm{1}+\frac{{x}−\mathrm{4049}}{\mathrm{2026}}−\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow\frac{{x}−\mathrm{6075}}{\mathrm{2024}}+\frac{{x}−\mathrm{6075}}{\mathrm{2025}}\:+\frac{{x}−\mathrm{6075}}{\mathrm{2026}}=\mathrm{0} \\ $$$$\Rightarrow\left({x}−\mathrm{6075}\right)\:=\mathrm{0} \\…
Question Number 217149 by Abdullahrussell last updated on 03/Mar/25 Answered by vnm last updated on 04/Mar/25 $${there}\:{are}\:{at}\:{least}\:{two}\:{different}\:{solutions}\:{of}\:{this}\:{system}: \\ $$$$\left(\mathrm{1},\mathrm{2},\mathrm{3},−\mathrm{2}\right); \\ $$$$\approx\left(−\mathrm{0}.\mathrm{871039468742},\:\mathrm{2}.\mathrm{026118074551},\:\mathrm{2}.\mathrm{997458794026},\:−\mathrm{1}.\mathrm{995175324811}\right) \\ $$ Terms of…
Question Number 217163 by hardmath last updated on 03/Mar/25 $$\mathrm{If}\:\:\:\mathrm{a}+\mathrm{b}=\mathrm{b}+\mathrm{c}=\mathrm{4} \\ $$$$\mathrm{find}:\:\:\:\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} −\mathrm{8c}\:=\:? \\ $$ Answered by AntonCWX08 last updated on 03/Mar/25 $${a}+{b}={b}+{c}\Rightarrow{a}={c} \\…