Question Number 110419 by Aina Samuel Temidayo last updated on 28/Aug/20 $$\mathrm{Let}\:\mathrm{t}\:\mathrm{be}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} −\mathrm{3x}+\mathrm{1}=\mathrm{0},\:\mathrm{if}\: \\ $$$$\frac{\mathrm{t}^{\mathrm{2}} +\mathrm{pt}+\mathrm{1}}{\mathrm{t}^{\mathrm{2}} −\mathrm{t}+\mathrm{1}}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{t}+\mathrm{c}\:\mathrm{for} \\ $$$$\mathrm{some}\:\mathrm{p},\mathrm{c}\:\in\:\mathbb{Z},\:\mathrm{then}\:\mathrm{p}−\mathrm{c}\:\mathrm{equals}? \\ $$ Commented by Her_Majesty last…
Question Number 175951 by blackmamba last updated on 10/Sep/22 $$\:{Given}\:{f}\left({x}\right)={ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}\: \\ $$$$\:{a}\neq\:\mathrm{0}\:{and}\:\mid{f}\:'\left({x}\right)\mid\:\leqslant\:\mathrm{1}\:{for}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\: \\ $$$$\:{find}\:{max}\:{value}\:{of}\:{a}. \\ $$ Commented by infinityaction last updated on 10/Sep/22…
Question Number 44876 by Tinkutara last updated on 05/Oct/18 Answered by ajfour last updated on 06/Oct/18 $${let}\:\:{z}_{\mathrm{1}} =\:{x}\:\:,\:\:{z}_{\mathrm{2}} \:=\:{y}\:\:\:\:\left({just}\:{calling}\right) \\ $$$$\:\:\:\:\begin{cases}{\:\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{3}\boldsymbol{{y}}^{\mathrm{2}} \right)=\mathrm{2}}\\{\:\boldsymbol{{y}}\left(\mathrm{3}\boldsymbol{{x}}^{\mathrm{2}} −\boldsymbol{{y}}^{\mathrm{2}} \right)=\mathrm{1}}\end{cases}…
Question Number 175943 by CrispyXYZ last updated on 10/Sep/22 $$\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…