Question Number 139695 by john_santu last updated on 30/Apr/21 $$\:\mid{x}^{\mathrm{2}} −\mathrm{4}{x}\mid\:+\mid{x}\mid\:\geqslant\:\mathrm{3}\: \\ $$ Answered by MJS_new last updated on 30/Apr/21 $$\mid{x}\left({x}−\mathrm{4}\right)\mid+\mid{x}\mid=\mathrm{3} \\ $$$$\mathrm{squaring} \\ $$$${x}^{\mathrm{2}}…
Question Number 74148 by mathocean1 last updated on 19/Nov/19 $$\begin{cases}{−{x}\sqrt{\mathrm{3}}+\mathrm{2}{my}\sqrt{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}}\\{\mathrm{2}{mx}−\mathrm{3}{y}\sqrt{\mathrm{6}}=\mathrm{1}}\end{cases}\:\:\:\:\:\: \\ $$$$ \\ $$$${help}\:{me}\:{solve}\:{it}. \\ $$ Answered by MJS last updated on 19/Nov/19 $$\mathrm{2}\:\mathrm{equations}\:\mathrm{in}\:\mathrm{3}\:\mathrm{unknowns}\:\Rightarrow\:\mathrm{parametric} \\…
Question Number 139672 by mathlove last updated on 30/Apr/21 Answered by MJS_new last updated on 30/Apr/21 $$\left({x}^{{a}} \right)^{{b}} ={x}^{{ab}} \\ $$$$\left({x}^{{a}} \right)^{{a}} ={x}^{{a}^{\mathrm{2}} } \\…
Question Number 8602 by tawakalitu last updated on 17/Oct/16 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{that}\:\mathrm{will}\:\mathrm{make} \\ $$$$\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{13x}^{\mathrm{3}} \:+\:\mathrm{6x}^{\mathrm{2}} \:+\:\mathrm{px}\:+\:\mathrm{q}\:\: \\ $$$$\mathrm{a}\:\mathrm{perfect}\:\mathrm{square} \\ $$ Answered by sandy_suhendra last updated on…
Question Number 8595 by diofanto last updated on 17/Oct/16 $${Let}\:{N}\:\in\:\mathbb{Z}.\:{Show}\:{that} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{N}−\mathrm{1}} {\sum}}\frac{{N}−{k}}{{sin}\left({k}\pi/{N}\right)}\:>\:\mathrm{2}{N}−\mathrm{2}+\underset{{k}=\mathrm{2}} {\overset{{N}−\mathrm{1}} {\sum}}\frac{{N}−{k}}{{sin}\left(\left({k}−\mathrm{1}\right)\pi/\left({N}−\mathrm{1}\right)\right)} \\ $$$${if},\:{and}\:{only}\:{if},\:{N}\:\geqslant\:\mathrm{12} \\ $$ Commented by prakash jain last…
Question Number 74129 by malwaan last updated on 19/Nov/19 $$\mathrm{2}\boldsymbol{{C}}_{\mathrm{4}} ^{\boldsymbol{{n}}} \:=\:\mathrm{35}\boldsymbol{{C}}_{\mathrm{3}} ^{\frac{\boldsymbol{{n}}}{\mathrm{2}}} \: \\ $$$$\Rightarrow\:\boldsymbol{{n}}\:=\:? \\ $$ Answered by MJS last updated on 20/Nov/19…
Question Number 74121 by Raxreedoroid last updated on 19/Nov/19 $$\mathrm{Factor}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\left(\frac{{c}}{\mathrm{2}}\right){x}^{\mathrm{2}} +\left({b}−\frac{\mathrm{3}{c}}{\mathrm{2}}\right){x}+\left({c}−{b}+{a}\right) \\ $$ Commented by MJS last updated on 19/Nov/19 $$\left({x}+\frac{{b}}{{c}}−\frac{\mathrm{3}}{\mathrm{2}}+\frac{\sqrt{\left(\mathrm{2}{b}−{c}\right)^{\mathrm{2}} −\mathrm{8}{ac}}}{\mathrm{2}{c}}\right)\left({x}+\frac{{b}}{{c}}−\frac{\mathrm{3}}{\mathrm{2}}−\frac{\sqrt{\left(\mathrm{2}{b}−{c}\right)^{\mathrm{2}} −\mathrm{8}{ac}}}{\mathrm{2}{c}}\right)…
Question Number 8585 by tawakalitu last updated on 17/Oct/16 Commented by ridwan balatif last updated on 17/Oct/16 $$\mathrm{we}\:\mathrm{must}\:\mathrm{count}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}? \\ $$ Commented by ridwan balatif last…
Question Number 74111 by FCB last updated on 19/Nov/19 Commented by MJS last updated on 20/Nov/19 $$\mathrm{no}\:“\mathrm{beautiful}''\:\mathrm{solution} \\ $$ Answered by mind is power last…
Question Number 139644 by mathdanisur last updated on 30/Apr/21 $${a};{b};{c}\in\mathbb{R},\:\mid{ax}^{\mathrm{5}} +{bx}^{\mathrm{3}} +{cx}\mid\leqslant\mathrm{1},\:\forall\mid{x}\mid\leqslant\mathrm{1} \\ $$$${proof}:\:\mid{a}\mid\leqslant\mathrm{16},\:\mid{b}\mid\leqslant\mathrm{20},\:\mid{c}\mid\leqslant\mathrm{5} \\ $$ Answered by ajfour last updated on 30/Apr/21 $${let}\:\:\mid{x}\mid={r} \\…