Question Number 36019 by math1967 last updated on 27/May/18 $${If}\:{a}+{b}+{c}=\mathrm{0}\:{show}\:{that} \\ $$$$\left(\frac{{a}}{{b}−{c}}+\frac{{b}}{{c}−{a}}+\frac{{c}}{{a}−{b}}\right)\left(\frac{{b}−{c}}{{a}}+\frac{{c}−{a}}{{b}}\:+\frac{{a}−{b}}{{c}}\right)=\mathrm{9} \\ $$ Answered by ajfour last updated on 27/May/18 $${let}\:{b}={pa}\:\:,\:\:{c}={qa} \\ $$$$\Rightarrow\:{a}\left(\mathrm{1}+{p}+{q}\right)=\mathrm{0}\:\:\:\:\:\:\:\:\:\:…\left({i}\right) \\…
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Question Number 167086 by mathlove last updated on 06/Mar/22 Commented by cortano1 last updated on 06/Mar/22 $$\:\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)=\frac{\mathrm{x}^{\mathrm{2010}} }{\mathrm{x}^{\mathrm{2010}} +\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2010}} }+\frac{\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2010}} }{\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2010}} +\mathrm{x}^{\mathrm{2010}} }=\mathrm{1} \\ $$$$=\:\mathrm{1005}…
Question Number 167066 by mathlove last updated on 05/Mar/22 Commented by cortano1 last updated on 05/Mar/22 $$\:\mathrm{x}\sqrt{\mathrm{x}}\:=\:\mathrm{u}\Rightarrow\mathrm{x}=\mathrm{u}^{\frac{\mathrm{2}}{\mathrm{3}}} \\ $$$$\mathrm{f}\left(\mathrm{u}\right)=\frac{\mathrm{u}^{\frac{\mathrm{1}}{\mathrm{3}}} }{\mathrm{u}^{\frac{\mathrm{2}}{\mathrm{3}}} }\:=\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{u}}} \\ $$$$\mathrm{f}\left(−\mathrm{64}\right)+\mathrm{f}\left(−\mathrm{8}\right)+\mathrm{f}\left(\mathrm{1}\right)+\mathrm{f}\left(\mathrm{27}\right)= \\ $$$$\:\frac{\mathrm{1}}{−\mathrm{4}}+\frac{\mathrm{1}}{−\mathrm{2}}+\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{3}}…
Question Number 101514 by bemath last updated on 03/Jul/20 Commented by Dwaipayan Shikari last updated on 03/Jul/20 $$\left({a}+{b}+{c}\right)^{\mathrm{2}} =\mathrm{9} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{2}{bc}+\mathrm{2}{ac}=\mathrm{9} \\…
Question Number 167036 by HongKing last updated on 05/Mar/22 $$\mathrm{Evaluate}: \\ $$$$\Omega\:=\:\int\:\mathrm{620}\:\left(\mathrm{x}^{\mathrm{2017}} \:-\:\mathrm{69}\:\mathrm{x}^{\mathrm{126}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$ Answered by mr W last updated on 05/Mar/22…
Question Number 167037 by HongKing last updated on 05/Mar/22 $$\mathrm{If}\:\:\:\:\:\mathrm{I}\:=\:\int_{-\mathrm{1}} ^{\:-\mathrm{8}} \:\frac{\mathrm{1}}{\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \:−\:\mathrm{x}}\:\mathrm{dx} \\ $$$$\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\mathrm{81}\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{I}}} \:? \\ $$ Answered by ghakhan last updated on 05/Mar/22…
Question Number 101498 by bemath last updated on 03/Jul/20 $$\mathrm{If}\:\mathrm{4}^{\mathrm{x}} \:=\:\mathrm{5}^{\mathrm{y}} \:=\:\mathrm{20}^{\mathrm{z}} \:. \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{z}\:\mathrm{in}\:\mathrm{term}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}. \\ $$$$\bigstar\spadesuit \\ $$ Commented by bobhans last updated on…
Question Number 167038 by HongKing last updated on 05/Mar/22 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{a}\:\:\mathrm{and}\:\:\mathrm{b} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{y}=\mathrm{0},\mathrm{5x}^{\mathrm{2}} +\left(\mathrm{a}+\mathrm{b}\right)\mathrm{x}+\mathrm{ab} \\ $$$$\mathrm{is}\:\mathrm{always}\:\mathrm{positive} \\ $$ Commented by mr W last updated on 05/Mar/22…
Question Number 101489 by bramlex last updated on 03/Jul/20 Answered by john santu last updated on 03/Jul/20 $$\mathrm{let}\:\sqrt[{\mathrm{3}\:}]{\mathrm{x}−\mathrm{3}}\:=\:{h}\:\mathrm{and}\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}+\mathrm{4}}\:=\:{t} \\ $$$$\Leftrightarrow\:\mathrm{5}{ht}\:+\:\mathrm{4}{h}^{\mathrm{2}} \:+\:{t}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\left(\mathrm{4}{h}+{t}\right)\left({h}+{t}\right)\:=\:\mathrm{0} \\…