Question Number 101800 by Quvonchbek last updated on 04/Jul/20 Commented by mr W last updated on 05/Jul/20 $${very}\:{interesting}!\:{thanks}\:{sir}! \\ $$ Commented by mr W last…
Question Number 36259 by behi83417@gmail.com last updated on 30/May/18 Answered by ajfour last updated on 30/May/18 $${let}\:\:\:\:{a}+{b}={c}+{d}={s}\:\:\:\:\:\:\:\:\:\:\:…\:\left({i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:{cd}\left({ab}−\mathrm{1}\right)={ab}\:\:\:\:\:\:\:\:\:\:…\left({ii}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\frac{{ad}+{bc}}{{bd}}=\frac{{a}+{c}}{{b}+{d}}\:\:\:\:\:\:\:\:\:\:\:\:\:….\left({iii}\right) \\ $$$$\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{{ab}}=\frac{{c}^{\mathrm{2}}…
Question Number 167324 by mathlove last updated on 13/Mar/22 Answered by Rasheed.Sindhi last updated on 13/Mar/22 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{9}}+…+\frac{\mathrm{1}}{\mathrm{3}^{{n}} }}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{25}}+…+\frac{\mathrm{1}}{\mathrm{5}^{{n}} }}\right) \\ $$$$\:\:\:\:\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{9}}+…}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{25}}+…}=\frac{\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}}}{\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}}}}=\frac{\frac{\mathrm{1}}{\mathrm{2}/\mathrm{3}}}{\frac{\mathrm{1}}{\mathrm{4}/\mathrm{5}}}=\frac{\frac{\mathrm{3}}{\mathrm{2}}}{\frac{\mathrm{5}}{\mathrm{4}}} \\ $$$$\:\:\:=\frac{\mathrm{3}}{\mathrm{2}}×\frac{\mathrm{4}}{\mathrm{5}}=\frac{\mathrm{6}}{\mathrm{5}} \\…
Question Number 36221 by ajfour last updated on 30/May/18 $${a}\left({b}+\frac{\mathrm{1}}{{b}}\right)=−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:….\left({i}\right) \\ $$$${b}−\frac{\mathrm{1}}{{b}}={a}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\left({ii}\right) \\ $$$${Solve}\:{simultaneously}\:{for}\:{a},\:{and}\:{b}. \\ $$ Commented by behi83417@gmail.com last updated on 30/May/18 $${a}^{\mathrm{2}}…
Question Number 36219 by MJS last updated on 30/May/18 $$\mathrm{in}\:\mathrm{question}\:\mathrm{34992}\:\mathrm{this}\:\mathrm{had}\:\mathrm{to}\:\mathrm{be}\:\mathrm{solved}: \\ $$$${t}^{\mathrm{4}} +\mathrm{8}{t}^{\mathrm{3}} +\mathrm{2}{t}^{\mathrm{2}} −\mathrm{8}{t}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{here}\:\mathrm{another}\:\mathrm{concept}\:\mathrm{worked}\:\left(\mathrm{I}\:\mathrm{tried}\:\mathrm{some}\right) \\ $$$${t}_{\mathrm{1}} ={a}+{b}+{c}+{d} \\ $$$${t}_{\mathrm{2}} ={a}+{b}−{c}−{d} \\ $$$${t}_{\mathrm{3}}…
Question Number 36207 by MJS last updated on 30/May/18 $${x}^{\mathrm{4}} +\mathrm{10}{x}^{\mathrm{3}} +\mathrm{6}{x}−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{for}\:\mathrm{those}\:\mathrm{who}\:\mathrm{want}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{solution}… \\ $$$$ \\ $$$$\left({x}−{a}−{b}\mathrm{i}\right)\left({x}−{a}+{b}\mathrm{i}\right)\left({x}−{c}−{d}\right)\left({x}−{c}+{d}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{4}} −\mathrm{2}\left({a}+{c}\right){x}^{\mathrm{3}} +\left({a}^{\mathrm{2}} +\mathrm{4}{ac}+{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{d}^{\mathrm{2}}…
Question Number 101742 by bemath last updated on 04/Jul/20 $$\begin{cases}{{ab}+{a}+{b}\:=\:\mathrm{5}}\\{{bc}\:+\:{b}+{c}\:=\:\mathrm{14}}\\{{ac}\:+\:{a}+{c}\:=\:\mathrm{9}}\end{cases} \\ $$$$\mathrm{find}\:{a}+{b}+{c}\:=\:\_\_\_ \\ $$ Answered by bramlex last updated on 04/Jul/20 $$\left(\mathrm{1}\right)\:{a}\:=\:\frac{\mathrm{5}−{b}}{{b}+\mathrm{1}}\:=\:\frac{\mathrm{9}−{c}}{{c}+\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:{b}\:=\:\frac{\mathrm{14}−{c}}{{c}+\mathrm{1}}\: \\…
Question Number 167278 by mnjuly1970 last updated on 11/Mar/22 Answered by mr W last updated on 11/Mar/22 $${f}\left({x}\right)=\frac{\sqrt{{x}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}}{{x}^{\mathrm{2}} +\mathrm{1}+{x}}=\frac{\mathrm{1}}{\:\sqrt{\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}}}+\sqrt{\frac{{x}}{{x}^{\mathrm{2}} +\mathrm{1}}}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{{x}+\frac{\mathrm{1}}{{x}}}+\frac{\mathrm{1}}{\:\sqrt{{x}+\frac{\mathrm{1}}{{x}}}}} \\…
Question Number 167268 by mathlove last updated on 11/Mar/22 $${x}^{\frac{\mathrm{2}}{\:\sqrt[{\mathrm{5}}]{\mathrm{2}}}} =\mathrm{25}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}=? \\ $$ Commented by mkam last updated on 11/Mar/22 $${x}^{\mathrm{2}^{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}}} } =\:\mathrm{25} \\ $$$$…
Question Number 167248 by mathlove last updated on 10/Mar/22 $$\sqrt{\mathrm{3}^{{x}} }+\mathrm{1}=\mathrm{2}^{{x}} \:\:\:\:{faind}\:{x}=? \\ $$ Commented by mr W last updated on 10/Mar/22 $$\mathrm{3}^{\frac{{x}}{\mathrm{2}}} +\mathrm{1}=\mathrm{2}^{{x}} \\…