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Category: Algebra

solve-for-real-x-3-x-5-5x-4-10x-3-10x-2-5x-x-2-2x-1-x-4-4x-2-4-3-

Question Number 218482 by Nicholas666 last updated on 10/Apr/25 $$ \\ $$$$\:\:\:\:{solve}\:{for}\:{real}\:\boldsymbol{{x}}; \\ $$$$ \\ $$$$\underset{\:\:\:\mathrm{3}\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{5}} −\mathrm{5}\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{10}\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{10}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{x}}+\mathrm{1}}+\sqrt{\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}}}=\sqrt{\mathrm{3}\:}\:\:\:} {\:} \\…

Solve-the-following-equation-for-the-real-x-value-x-4-4x-3-6x-2-4x-2-2x-4-8x-3-12x-2-8x-5-

Question Number 218473 by Nicholas666 last updated on 10/Apr/25 $$ \\ $$$$\:{Solve}\:{the}\:{following}\:{equation}\:{for}\:{the}\:{real}\:\boldsymbol{{x}}\:{value}; \\ $$$$\:{x}^{\mathrm{4}} −\mathrm{4}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{2}\:=\:\sqrt{\mathrm{2}{x}^{\mathrm{4}} −\mathrm{8}{x}^{\mathrm{3}} +\mathrm{12}{x}^{\mathrm{2}} −\mathrm{8}{x}+\mathrm{5}}\:\:\:\: \\ $$$$ \\ $$ Answered…

Let-a-gt-1-fixed-Solve-for-real-numbers-the-system-a-x-2-x-ay-2-a-y-2-y-az-2-a-z-2-z-ax-2-

Question Number 218504 by hardmath last updated on 10/Apr/25 $$\mathrm{Let}\:\:\:\mathrm{a}>\mathrm{1}\:\:\:\mathrm{fixed} \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{system} \\ $$$$\begin{cases}{\mathrm{a}^{\boldsymbol{\mathrm{x}}} \:\:+\:\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:\:=\:\:\mathrm{ay}\:+\:\mathrm{2}}\\{\mathrm{a}^{\boldsymbol{\mathrm{y}}} \:\:+\:\:\mathrm{2}^{\boldsymbol{\mathrm{y}}} \:\:=\:\:\mathrm{az}\:\:+\:\:\mathrm{2}}\\{\mathrm{a}^{\boldsymbol{\mathrm{z}}} \:\:+\:\:\mathrm{2}^{\boldsymbol{\mathrm{z}}} \:\:=\:\:\mathrm{ax}\:\:+\:\:\mathrm{2}}\end{cases}\: \\ $$ Answered by vnm…

Question-218428

Question Number 218428 by hardmath last updated on 09/Apr/25 Answered by A5T last updated on 09/Apr/25 $$\mathrm{Let}\:\mathrm{F}\:\in\:\mathrm{BC}\:\mathrm{such}\:\mathrm{that}\:\mathrm{AF}\bot\mathrm{BC} \\ $$$$\mathrm{AB}=\mathrm{AE}\Rightarrow\mathrm{BF}=\mathrm{FE}=\frac{\mathrm{7}+\mathrm{3}}{\mathrm{2}}=\mathrm{5} \\ $$$$\mathrm{BF}×\mathrm{BC}=\mathrm{BA}^{\mathrm{2}} \Rightarrow\mathrm{5}\left(\mathrm{10}+\mathrm{x}\right)=\mathrm{BA}^{\mathrm{2}} …\left(\mathrm{i}\right) \\ $$$$\mathrm{AB}^{\mathrm{2}}…

Let-gt-0-fixed-Solve-for-real-numbers-the-system-x-2-yz-2-y-2-zx-7-2-z-2-xy-5-2-

Question Number 218364 by hardmath last updated on 08/Apr/25 $$\mathrm{Let}\:\:\:\boldsymbol{\lambda}>\mathrm{0}\:\:\:\mathrm{fixed} \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{system}:\:\:\:\begin{cases}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{yz}\:=\:\lambda^{\mathrm{2}} }\\{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{zx}\:=\:\mathrm{7}\lambda^{\mathrm{2}} }\\{\mathrm{z}^{\mathrm{2}} \:−\:\mathrm{xy}\:=\:−\mathrm{5}\lambda^{\mathrm{2}} }\end{cases} \\ $$ Answered by vnm last updated…

Find-lim-n-k-1-n-i-1-k-i-k-i-1-3-1-

Question Number 218365 by hardmath last updated on 08/Apr/25 $$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\left[\:\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\:\mathrm{i}\:\left(\mathrm{k}\:−\:\mathrm{i}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\right)\right]^{−\mathrm{1}} =\:? \\ $$ Answered by vnm last updated on 08/Apr/25…