Question Number 175943 by CrispyXYZ last updated on 10/Sep/22 $$\left(\mathrm{1}\right)\:\exists{x}\in\mathbb{R},\:{x}^{\mathrm{2}} +{x}+{a}=\mathrm{0}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{a}. \\ $$$$\left(\mathrm{2}\right)\:\forall{x}\in\mathbb{R},\:{x}^{\mathrm{2}} +{x}+{a}=\mathrm{0}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{a}. \\ $$ Answered by mahdipoor last updated on 10/Sep/22 $$\left.\mathrm{1}\right)\:{x}=\frac{−\mathrm{1}\pm\sqrt{\mathrm{1}−\mathrm{4}{a}}}{\mathrm{2}}\:\Rightarrow\:\mathrm{1}−\mathrm{4}{a}\geqslant\mathrm{0}\:\Rightarrow\:{a}\leqslant\frac{\mathrm{1}}{\mathrm{4}} \\…
Question Number 110399 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{The}\:\mathrm{identity} \\ $$$$\mathrm{2}\left[\mathrm{16a}^{\mathrm{4}} +\mathrm{81b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \right]=\left[\mathrm{4a}^{\mathrm{2}} +\mathrm{9b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} \right]^{\mathrm{2}} \\ $$$$\mathrm{cannot}\:\mathrm{result}\:\mathrm{from}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{equations}?\: \\…
Question Number 175916 by mnjuly1970 last updated on 09/Sep/22 $$ \\ $$$$\:\:\:\:{f}\left({x}\right)=\:\mathrm{2}^{\:\sqrt[{\mathrm{3}}]{\:{sin}\left({x}\right)\:+\sqrt{\mathrm{3}}\:{cos}\left({x}\right)}\:} −\:\mathrm{2}^{\:\sqrt[{\mathrm{3}}]{−{sin}\left({x}\right)\:−\sqrt{\mathrm{3}}\:{cos}\left({x}\right)}} \\ $$$$\:\:\:\:\:\:\:{R}_{\:{f}} \:=? \\ $$ Commented by mnjuly1970 last updated on 09/Sep/22…
Question Number 110374 by Aina Samuel Temidayo last updated on 28/Aug/20 $$\mathrm{If}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{whose}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{coefficients}\:\mathrm{is}\:\mathrm{3}\:\mathrm{and}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{factorised}\:\mathrm{into}\:\mathrm{two}\:\mathrm{polynomials} \\ $$$$\mathrm{Q}\left(\mathrm{x}\right),\mathrm{R}\left(\mathrm{x}\right)\:\mathrm{with}\:\mathrm{integer}\:\mathrm{coefficients}, \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients} \\ $$$$\mathrm{Q}\left(\mathrm{x}\right)^{\mathrm{2}} +\mathrm{R}\left(\mathrm{x}\right)^{\mathrm{2}} \:\mathrm{is} \\…
Question Number 175893 by blackmamba last updated on 08/Sep/22 $$\:{If}\:{outer}\:{angle}\:{in}\:{n}−{polygone}\:{is}\:\mathrm{18}° \\ $$$$\:{find}\:{n} \\ $$ Commented by Rasheed.Sindhi last updated on 09/Sep/22 $${n}=\mathrm{20} \\ $$ Commented…
Question Number 110359 by Aina Samuel Temidayo last updated on 28/Aug/20 $$\mathrm{Let} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mid\mathrm{x}−\mathrm{2}\mid+\mid\mathrm{x}−\mathrm{4}\mid−\mid\mathrm{2x}−\mathrm{6}\mid, \\ $$$$\mathrm{for}\:\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{8}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{largest}\:\mathrm{and} \\ $$$$\mathrm{smallest}\:\mathrm{values}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is} \\ $$ Answered by bobhans…
Question Number 44819 by MrW3 last updated on 05/Oct/18 Commented by MrW3 last updated on 05/Oct/18 $${see}\:{also}\:{Q}\mathrm{19198} \\ $$ Commented by MrW3 last updated on…
Question Number 175871 by infinityaction last updated on 08/Sep/22 $$\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{4x}+\mathrm{13}}\:+\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{14x}+\mathrm{130}} \\ $$$$\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{f}\left(\mathrm{x}\right)\:\:\mathrm{x}\:\in\:\mathbb{R}\: \\ $$ Answered by mr W last updated on 08/Sep/22 $${f}\left({x}\right)=\sqrt{\left({x}−\mathrm{2}\right)^{\mathrm{2}}…
Question Number 175860 by Shrinava last updated on 08/Sep/22 Answered by mr W last updated on 08/Sep/22 $${we}\:{know}: \\ $$$$\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{2}} }+…=\frac{\pi^{\mathrm{2}} }{\mathrm{6}}…
Question Number 44783 by Necxx last updated on 04/Oct/18 $$\left({x}^{\mathrm{2}} \right)^{\mathrm{2}} +\left({y}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{97}…….\mathrm{1} \\ $$$$\left({x}^{\mathrm{2}} \right)^{\mathrm{3}} +\left({y}^{\mathrm{2}} \right)^{\mathrm{3}} =\mathrm{793}…….\mathrm{2} \\ $$$${solve}\:{the}\:{simultaneous}\:{equation} \\ $$ Answered…