Category: Algebra

Question Number 50908 by Smail last updated on 22/Dec/18 $$Givenf\left(x\right)=\underset{k=0}{\sum ^n}{}^n{C}_{k}sin\left(kx\right)cos\left((n-k)x\right)$$$$Findasimpleformforf\left(x\right)$$$$(Youranswershouldbewrittenlikec\left(n\right).g\left(nx\right))$$ Answered by Smail last updated…

Question Number 181923 by manxsol last updated on 02/Dec/22 Answered by SEKRET last updated on 02/Dec/22 $$\mathbf{xlog}2\left(\mathbf{y}\right)=6\mathbf{y}=2^{\frac{6}{\mathbf{x}}}$$$$\left(\frac{\mathbf{x}+\mathbf{y}}{\mathbf{x}-\mathbf{y}}\right)\mathbf{log}2\left(\mathbf{y}\right)=4$$$$\frac{6}{\mathbf{x}}=\frac{4\mathbf{x}-4\mathbf{y}}{\mathbf{x}+\mathbf{y}}6\mathbf{x}+6\mathbf{y}=4{\mathbf{x}}^2-4\mathbf{xy}$$$$\:\:\boldsymbol{\mathrm{y}}\left(\mathrm{6}+\mathrm{4}\boldsymbol{\mathrm{x}}\right)=\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{6}\boldsymbol{\mathrm{x}}\:\:\:\:\:\boldsymbol{\mathrm{y}}=\:\frac{\boldsymbol{\mathrm{x}}\left(\mathrm{2}\boldsymbol{\mathrm{x}}−\mathrm{3}\right)}{\left(\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{3}\right)}…

Question Number 181902 by Acem last updated on 01/Dec/22 Answered by Rasheed.Sindhi last updated on 02/Dec/22 $$(x^2+xy+y^2)\sqrt{x^2+y^2}=185\dots .\left(i\right)$$$$\left({x}^{\mathrm{2}} −{xy}+{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}}…

Question Number 50825 by behi83417@gmail.com last updated on 20/Dec/18 $${\mathbf{x}}^4={\mathbf{ax}}^2+{\mathbf{by}}^2$$$${\mathbf{y}}^4={\mathbf{bx}}^2+{\mathbf{ay}}^2$$$$\mathbf{solve}\mathbf{for}\mathbf{x},\mathbf{y}.[\mathbf{a},\mathbf{b}\in \mathbf{R};\mathbf{a},\mathbf{b}\ne 0]$$ Answered by mr W last…

Question Number 116323 by bobhans last updated on 03/Oct/20 $$\left(1\right)\mathrm{Let}\mathrm{a},\mathrm{b}\mathrm{and}\mathrm{c}\mathrm{real}\mathrm{number}\mathrm{such}\mathrm{that}$$$$\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}=\frac{1}{3},\frac{\mathrm{bc}}{\mathrm{b}+\mathrm{c}}=\frac{1}{4}\mathrm{and}\frac{\mathrm{ac}}{\mathrm{a}+\mathrm{c}}=\frac{1}{5}.\mathrm{Find}$$$$\mathrm{the}\mathrm{value}\mathrm{of}\frac{24\mathrm{abc}}{\mathrm{ab}+\mathrm{ac}+\mathrm{bc}}?$$$$\left(2\right)\mathrm{Let}\mathrm{p}\mathrm{and}\mathrm{q}\mathrm{be}\mathrm{two}\mathrm{real}\mathrm{number}\mathrm{that}$$$$\mathrm{satisfy}\mathrm{p}.\mathrm{q}=2013.\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{minimum}$$$$\mathrm{value}\mathrm{of}{(\mathrm{p}+\mathrm{q})}^2?$$ Answered by…