Question Number 127924 by ajfour last updated on 03/Jan/21 $$\:\:\:\:{x}^{\mathrm{3}} ={x}+{c} \\ $$$${Solve}\:{for}\:{x},\:{even}\:{when}\:{c}<\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}}. \\ $$ Commented by ajfour last updated on 03/Jan/21 $$\left({without}\:{trigonometric}\:{functions}\right) \\ $$…
Question Number 62372 by Rasheed.Sindhi last updated on 20/Jun/19 $${Solve}\:{for}\:{x}\:,\:{y} \\ $$$$\mathrm{3}{x}>\mathrm{2}{y}\:\wedge\:\mathrm{2}{x}<\mathrm{3}{y}\: \\ $$$${where}\:{x},{y}\in\mathbb{N} \\ $$ Answered by MJS last updated on 20/Jun/19 $$\frac{\mathrm{2}}{\mathrm{3}}{y}<{x}<\frac{\mathrm{3}}{\mathrm{2}}{y}\:\wedge\:{y}>\mathrm{0} \\…
Question Number 127908 by bramlexs22 last updated on 03/Jan/21 $$\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{1}.\mathrm{3}.\mathrm{5}}+\frac{\mathrm{3}}{\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}}+\frac{\mathrm{4}}{\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}.\mathrm{9}}+…=? \\ $$ Answered by liberty last updated on 03/Jan/21 $$\:\mathrm{Let}\:\lambda\:=\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{n}}{\mathrm{1}.\mathrm{3}.\mathrm{5}…\left(\mathrm{2n}+\mathrm{1}\right)}\: \\ $$$$\mathrm{consider}\:\frac{\mathrm{n}}{\mathrm{1}.\mathrm{3}.\mathrm{5}….\left(\mathrm{2n}+\mathrm{1}\right)}\:=\:\frac{\mathrm{n}}{\mathrm{1}.\left(\mathrm{3}.\mathrm{5}…\left(\mathrm{2n}−\mathrm{1}\right)\right).\left(\mathrm{2n}+\mathrm{1}\right)} \\…
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Question Number 62334 by smartsmith459@gmail.com last updated on 19/Jun/19 $${if}\:\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\:\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{2}\alpha\beta\:{evaluate}\left(\alpha−\beta\right) \\ $$ Answered by Kunal12588 last updated on 20/Jun/19 $$\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\left(\alpha+\beta\right)^{\mathrm{2}}…
Question Number 62281 by behi83417@gmail.com last updated on 19/Jun/19 $$\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{y}}^{\mathrm{3}} =\mathrm{3}\boldsymbol{\mathrm{xy}}}\\{\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\boldsymbol{\mathrm{y}}^{\mathrm{4}} =\mathrm{4}\boldsymbol{\mathrm{xy}}}\end{cases}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\neq\mathrm{0}\right] \\ $$ Answered by mr W last updated on 19/Jun/19 $${let}\:{u}={x}+{y},\:{v}={xy}…
Question Number 62276 by behi83417@gmail.com last updated on 18/Jun/19 $$\begin{cases}{\frac{\sqrt{\boldsymbol{\mathrm{x}}}}{\boldsymbol{\mathrm{a}}}+\frac{\sqrt{\boldsymbol{\mathrm{y}}}}{\boldsymbol{\mathrm{b}}}=\mathrm{1}}\\{\frac{\sqrt{\boldsymbol{\mathrm{a}}}}{\boldsymbol{\mathrm{x}}}+\frac{\sqrt{\boldsymbol{\mathrm{b}}}}{\boldsymbol{\mathrm{y}}}=\mathrm{1}}\end{cases}\:\:\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\mathrm{R}^{+} \\ $$ Answered by mr W last updated on 19/Jun/19 $${let}\:{X}=\sqrt{{x}},\:{Y}=\sqrt{{y}},\:{A}=\sqrt{{a}},\:{B}=\sqrt{{b}} \\ $$$$\frac{{X}}{{A}^{\mathrm{2}} }+\frac{{Y}}{{B}^{\mathrm{2}} }=\mathrm{1}…
Question Number 62275 by behi83417@gmail.com last updated on 18/Jun/19 $$\begin{cases}{\boldsymbol{\mathrm{a}}\sqrt{\boldsymbol{\mathrm{x}}}+\boldsymbol{\mathrm{b}}\sqrt{\boldsymbol{\mathrm{y}}}=\mathrm{2}\sqrt{\boldsymbol{\mathrm{ab}}}}\\{\boldsymbol{\mathrm{x}}\sqrt{\boldsymbol{\mathrm{a}}}+\boldsymbol{\mathrm{y}}\sqrt{\boldsymbol{\mathrm{b}}}=\mathrm{2}\sqrt{\boldsymbol{\mathrm{ab}}}}\end{cases}\:\:\:\:\:\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\mathrm{R}^{+} \\ $$ Answered by mr W last updated on 19/Jun/19 $${let}\:{X}=\sqrt{{x}},{Y}=\sqrt{{y}},{A}=\sqrt{{a}},{B}=\sqrt{{b}} \\ $$$${A}^{\mathrm{2}} {X}+{B}^{\mathrm{2}} {Y}=\mathrm{2}{AB}…
Question Number 127788 by peter frank last updated on 02/Jan/21 Answered by mr W last updated on 02/Jan/21 $${AB}=\left[{ab}\right]=\mathrm{10}{a}+{b} \\ $$$${CD}=\left[{ba}\right]={a}+\mathrm{10}{b} \\ $$$${with}\:\mathrm{1}\leqslant{a},{b}\leqslant\mathrm{9} \\ $$$$…