Question Number 214499 by hardmath last updated on 10/Dec/24 $$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\left(\mathrm{x}\:\:+\:\:\mathrm{3y}\right)\:\:=\:\:\mathrm{11}}\\{\mathrm{y}^{\mathrm{2}} \:\:+\:\:\left(\mathrm{y}\:\:+\:\:\mathrm{3x}\right)\:\:=\:\:\mathrm{29}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\:\mathrm{x}\:+\:\mathrm{y}\:=\:? \\ $$ Commented by Frix last updated on 10/Dec/24 $$\mathrm{4}\:\mathrm{solutions} \\ $$$${x}+{y}={t} \\…
Question Number 214456 by hardmath last updated on 09/Dec/24 $$\mathrm{If}\:\:\:\frac{\mathrm{x}}{\mathrm{a}^{\mathrm{2}} −\:\mathrm{bc}}\:=\:\frac{\mathrm{y}}{\mathrm{b}^{\mathrm{2}} −\:\mathrm{ac}}\:=\:\frac{\mathrm{z}}{\mathrm{c}^{\mathrm{2}} −\:\mathrm{ab}} \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{ax}\:+\:\mathrm{by}\:+\:\mathrm{cz}}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}}\:=\:? \\ $$ Commented by Ghisom last updated on 09/Dec/24 $${a}+{b}+{c}…
Question Number 214457 by hardmath last updated on 09/Dec/24 $$\mathrm{If}\:\:\:\frac{\left(\mathrm{x}\:+\:\mathrm{2y}\:+\:\mathrm{3z}\right)^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} }\:=\:\mathrm{14}\:\:\:\:\:\mathrm{find}:\:\:\frac{\mathrm{x}\:+\:\mathrm{y}}{\mathrm{z}}\:=\:? \\ $$ Commented by Ghisom last updated on 09/Dec/24 $$\mathrm{1} \\…
Question Number 214454 by hardmath last updated on 09/Dec/24 $$\mathrm{a},\mathrm{b},\mathrm{c}\:\in\:\mathbb{R}^{+} \\ $$$$\mathrm{S}\:\:=\:\:\frac{\mathrm{9a}}{\mathrm{b}\:+\:\mathrm{c}}\:\:+\:\:\frac{\mathrm{16b}}{\mathrm{a}\:+\:\mathrm{c}}\:\:+\:\:\frac{\mathrm{49c}}{\mathrm{a}\:+\:\mathrm{b}} \\ $$$$\boldsymbol{\mathrm{min}}\left(\mathrm{S}\right)\:=\:? \\ $$ Commented by Ghisom last updated on 09/Dec/24 $${S}>\mathrm{24} \\…
Question Number 214455 by hardmath last updated on 09/Dec/24 $$\mathrm{If}\:\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{xyz} \\ $$$$\mathrm{Find}: \\ $$$$\frac{\mathrm{x}\left(\mathrm{1}−\mathrm{y}^{\mathrm{2}} \right)\left(\mathrm{1}−\mathrm{z}^{\mathrm{2}} \right)+\mathrm{y}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{1}−\mathrm{z}^{\mathrm{2}} \right)+\mathrm{z}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{1}−\mathrm{y}^{\mathrm{2}} \right)}{\mathrm{2xyz}} \\ $$ Commented by Ghisom…
Question Number 214443 by hardmath last updated on 08/Dec/24 $$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}\:\in\:\mathrm{Q} \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{2}}}\:=\:\mathrm{2}^{\boldsymbol{\mathrm{a}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{b}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{c}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{d}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{e}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{f}}} \\ $$$$\mathrm{find}:\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}\:=\:? \\ $$ Answered by ajfour last…
Question Number 214421 by ChantalYah last updated on 08/Dec/24 Answered by TonyCWX08 last updated on 08/Dec/24 $${Use}\:{Newton}'{s}\:{Identity} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 214419 by 2universe456 last updated on 08/Dec/24 Answered by golsendro last updated on 08/Dec/24 $$\:\:\:\:\begin{cases}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{2xy}=\mathrm{25}}\\{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{47}=\mathrm{10xy}}\end{cases}\Rightarrow\mathrm{2xy}+\mathrm{47}=\mathrm{10xy}+\mathrm{25} \\ $$$$\:\:\:\mathrm{8xy}=\:\mathrm{22}\Rightarrow\mathrm{4xy}=\mathrm{11} \\ $$$$\:\:\:\mathrm{4x}\left(\mathrm{5}−\mathrm{x}\right)=\mathrm{11}\:\Rightarrow\mathrm{4x}^{\mathrm{2}}…
Question Number 214414 by ChantalYah last updated on 07/Dec/24 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}=\mathrm{0}\:\mathrm{are}\:\alpha\:\mathrm{and}\:\beta, \\ $$$$\:\mathrm{show}\:\mathrm{that}; \\ $$$$\lambda\mu\mathrm{b}^{\mathrm{2}} =\mathrm{ac}\left(\lambda+\mu\right)^{\mathrm{2}} \:\mathrm{where}\:\frac{\alpha}{\beta}=\frac{\lambda}{\mu} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Mr}\:{Hans} \\ $$ Answered by…
Question Number 214408 by Abdullahrussell last updated on 07/Dec/24 Answered by A5T last updated on 07/Dec/24 $${Let}\:{f}\left({x}\right)={x}^{\mathrm{4}} −\mathrm{10}{x}^{\mathrm{3}} +\mathrm{37}{x}^{\mathrm{2}} −\mathrm{60}{x}+\mathrm{32} \\ $$$${f}\left(\mathrm{1}\right)={f}\left(\mathrm{4}\right)=\mathrm{0}\Rightarrow{f}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{4}\right)\left({x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{8}\right) \\…