Category: Algebra

Question Number 150531 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Find}\mathrm{x};\mathrm{y};\mathrm{x}\in \mathrm{Q}\mathrm{and}\mathrm{y}\in \mathrm{Z}\mathrm{such}\mathrm{that}:$$$$2020({\mathrm{x}}^2+{\mathrm{y}}^2)+2019(\mathrm{x}+\mathrm{y})=2021\mathrm{xy}$$ Commented by Rasheed.Sindhi last updated on 14/Aug/21 $$\underset{\smile} {\overset{\frown}…

Question Number 150515 by mathdanisur last updated on 13/Aug/21 Answered by ajfour last updated on 13/Aug/21 $$x=t+h$$$$7\sqrt{4t^2+8ht+4h^2+5t+5h-1}$$$$−\mathrm{7}\sqrt{\mathrm{4}{t}^{\mathrm{2}} +\mathrm{8}{ht}+\mathrm{4}{h}^{\mathrm{2}} −\mathrm{12}{t}−\mathrm{12}{h}+\mathrm{12}}…

Question Number 19434 by Tinkutara last updated on 11/Aug/17 $$\mathrm{If}{\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)}^n\mathrm{is}\mathrm{an}\mathrm{integer},\mathrm{then}n\mathrm{is}$$ Answered by ajfour last updated on 11/Aug/17 $$\left(\frac{\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{1}−{i}\sqrt{\mathrm{3}}}\right)^{\mathrm{n}} =\left(\frac{\mathrm{e}^{{i}\pi/\mathrm{6}} }{\mathrm{e}^{−{i}\pi/\mathrm{6}} }\right)^{\mathrm{n}} =\left[\mathrm{x}\right]…

Question Number 19433 by Tinkutara last updated on 11/Aug/17 $$\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}=$$ Answered by ajfour last updated on 11/Aug/17 $$=\frac{{(\sqrt{5+12\mathrm{i}}+\sqrt{5-12\mathrm{i}})}^2}{24\mathrm{i}}$$$$=\frac{10+2\sqrt{25+144}}{24\mathrm{i}}=\frac{3}{2\mathrm{i}}=-\frac{3}{2}\mathrm{i}.$$…

Question Number 19413 by Tinkutara last updated on 10/Aug/17 $$\mathrm{How}\mathrm{many}\mathrm{integer}\mathrm{pairs}(x,y)\mathrm{satisfy}$$$$x^2+4y^2-2xy-2x-4y-8=0?$$ Commented by mrW1 last updated on 11/Aug/17 $$\left(\mathrm{0},−\mathrm{1}\right) \…