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Category: Algebra

solve-x-2-5x-6-x-2-5x-4-x-2-x-2-5x-6-x-2-5x-4-x-2-2-x-2-4-find-x-

Question Number 171759 by Mikenice last updated on 20/Jun/22 $${solve}: \\ $$$$\left(\sqrt{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{6}}\:+\:\sqrt{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{4}}\:\right)^{\frac{{x}}{\mathrm{2}}} \:+\:\left(\sqrt{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{6}}\:−\:\sqrt{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{4}}\right)^{\frac{{x}}{\mathrm{2}}} \:=\:\mathrm{2}^{\frac{{x}+\mathrm{2}}{\mathrm{4}}} .\:{find}\:{x} \\ $$ Answered by Rasheed.Sindhi last…

Prove-that-in-any-triangle-ABC-with-area-F-holds-7-m-a-2-m-b-2-m-c-2-36F-3-cyc-a-m-a-2-

Question Number 171745 by Shrinava last updated on 20/Jun/22 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\mathrm{triangle}\:\:\mathrm{ABC}\:, \\ $$$$\mathrm{with}\:\mathrm{area}\:\:\mathrm{F}\:\:\mathrm{holds}: \\ $$$$\mathrm{7}\left(\mathrm{m}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{2}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{b}}} ^{\mathrm{2}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{2}} \right)\:\geqslant\:\mathrm{36F}\:+\:\mathrm{3}\:\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\left(\mathrm{a}−\mathrm{m}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{2}} \right) \\ $$…

solve-1-n-2-n-n-2-4n-

Question Number 171742 by Mikenice last updated on 20/Jun/22 $${solve}:\:\left(\mathrm{1}+{n}\right)!+\mathrm{2}\left({n}\right)!=\left({n}+\mathrm{2}\right)!−\mathrm{4}{n}! \\ $$ Commented by kaivan.ahmadi last updated on 20/Jun/22 $$\left({n}+\mathrm{2}\right)!−\left({n}+\mathrm{1}\right)!−\mathrm{6}{n}!=\mathrm{0} \\ $$$${n}!\left[\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)−\left({n}+\mathrm{1}\right)−\mathrm{6}\right]=\mathrm{0} \\ $$$$\Rightarrow{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}−{n}−\mathrm{1}−\mathrm{6}=\mathrm{0}\Rightarrow…

solve-12-x-2-4-x-find-x-

Question Number 171743 by Mikenice last updated on 20/Jun/22 $${solve}:\:\mathrm{12}^{{x}−\mathrm{2}} =\mathrm{4}^{{x}} ,\:{find}\:{x} \\ $$ Commented by kaivan.ahmadi last updated on 20/Jun/22 $$\mathrm{3}^{{x}−\mathrm{2}} ×\mathrm{4}^{{x}−\mathrm{2}} =\mathrm{4}^{{x}} \Rightarrow\mathrm{3}^{{x}−\mathrm{2}}…