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Category: Algebra

Question-194219

Question Number 194219 by Shlock last updated on 30/Jun/23 Answered by Frix last updated on 30/Jun/23 $$\left({x}^{\mathrm{3}} +\mathrm{4}{x}−\mathrm{8}\right)\left({x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{8}\right)\left({x}+\mathrm{2}\right)= \\ $$$$={x}^{\mathrm{7}} +\mathrm{64}{x}^{\mathrm{2}} −\mathrm{128} \\…

1-kx-1-3-x-1-has-two-real-roots-k-kx-1-1-3x-3x-2-x-3-x-3-3x-2-k-3-x-0-x-0-x-2-3x-k-3-0-1-0-9-4k-12-0-

Question Number 194209 by mnjuly1970 last updated on 01/Jul/23 $$ \\ $$$$\:\:\:\:\:\:\sqrt[{\mathrm{3}}]{\mathrm{1}+{kx}}\:+{x}\:=\:\mathrm{1}\:\:\:{has} \\ $$$$\:\:\:\:\:\:{two}\:{real}\:{roots}\:.\:\:\Rightarrow\:{k}=? \\ $$$$\:\:\:{kx}+\mathrm{1}=\:\mathrm{1}−\mathrm{3}{x}+\mathrm{3}{x}^{\mathrm{2}} −{x}^{\:\mathrm{3}} \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:{x}^{\:\mathrm{3}} \:−\mathrm{3}{x}^{\:\mathrm{2}} +\:\left({k}+\mathrm{3}\right){x}=\mathrm{0} \\ $$$$\:\:\:\:\:{x}=\mathrm{0}…

Question-194204

Question Number 194204 by Rupesh123 last updated on 30/Jun/23 Answered by MM42 last updated on 30/Jun/23 $${if}\:\:{n}={m}=\mathrm{0}\Rightarrow\mathrm{2}{a}_{\mathrm{0}} −\mathrm{1}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}{a}_{\mathrm{0}} \right)\Rightarrow{a}_{\mathrm{0}} =\mathrm{1} \\ $$$${if}\:\:{m}=\mathrm{1}\:,\:{n}=\mathrm{0}\Rightarrow\mathrm{2}{a}_{\mathrm{1}} −\mathrm{2}=\frac{\mathrm{1}}{\mathrm{2}}\left({a}_{\mathrm{2}} +{a}_{\mathrm{0}} \right)…

Question-194191

Question Number 194191 by JohnIsaac last updated on 29/Jun/23 Answered by Tinku Tara last updated on 30/Jun/23 $$\mathrm{4}{x}−\mathrm{3}{y}=\mathrm{9} \\ $$$$\mathrm{divide}\:\mathrm{both}\:\mathrm{sides}\:\mathrm{by}\:\mathrm{9}. \\ $$$$\frac{{x}}{\left(\mathrm{9}/\mathrm{4}\right)}+\frac{{y}}{\left(−\mathrm{3}\right)}=\mathrm{1} \\ $$$$\mathrm{x}−\mathrm{intercept}=\mathrm{9}/\mathrm{4} \\…

x-y-z-are-positive-real-numbers-if-x-4-y-4-z-4-1-Then-find-the-minimum-value-of-x-3-1-x-8-y-3-1-y-8-z-3-1-z-8-

Question Number 194105 by York12 last updated on 27/Jun/23 $$ \\ $$$${x}\:,\:{y}\:,\:{z}\:{are}\:{positive}\:{real}\:{numbers}\:{if}\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} =\mathrm{1} \\ $$$${Then}\:{find}\:{the}\:{minimum}\:{value}\:{of}\: \\ $$$$\frac{{x}^{\mathrm{3}} }{\mathrm{1}−{x}^{\mathrm{8}} }+\frac{{y}^{\mathrm{3}} }{\mathrm{1}−{y}^{\mathrm{8}} }+\frac{{z}^{\mathrm{3}} }{\mathrm{1}−{z}^{\mathrm{8}} }…