Question Number 139768 by Study last updated on 01/May/21
$${x}+\sqrt{{y}}=\mathrm{11} \\ $$$$\sqrt{{x}}+\sqrt{{y}}=\mathrm{7} \\ $$$${x}=? \\ $$$${y}=? \\ $$
Commented by Study last updated on 01/May/21
$${no}\:{type}\:{mater} \\ $$
Commented by Study last updated on 01/May/21
$${ok}\:{solve}\:{it} \\ $$
Answered by greg_ed last updated on 01/May/21
$$\blacksquare\:{x}\:=\:\mathrm{11}−\sqrt{{y}} \\ $$$$\blacksquare\:\sqrt{\mathrm{11}−\sqrt{{y}}}\:=\:\mathrm{7}−\sqrt{{y}}\:\Leftrightarrow \\ $$$$\:\:\:\:\:\mathrm{11}−\sqrt{{y}}\:=\:{y}\:−\mathrm{14}\sqrt{{y}}\:+\mathrm{49}\:\Leftrightarrow\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{13}\sqrt{{y}}\:=\:{y}\:+\mathrm{38}\:\Leftrightarrow\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{168}{y}\:\:=\:{y}^{\mathrm{2}} +\:\mathrm{76}{y}\:+\:\mathrm{1444}\:\Leftrightarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:{y}^{\mathrm{2}} \:−\mathrm{93}{y}\:+\:\mathrm{1444}\:=\:\mathrm{0}\:\Leftrightarrow \\ $$$$\mathrm{we}\:\mathrm{get}\:{y}\:=\:\frac{\mathrm{93}+\mathrm{13}\sqrt{\mathrm{17}}}{\mathrm{2}}\:\left(\boldsymbol{\mathrm{rejected}}\right)\:\&\:{y}\:=\:\frac{\mathrm{93}−\mathrm{13}\sqrt{\mathrm{17}}}{\mathrm{2}}\:\left(\boldsymbol{\mathrm{true}}\right) \\ $$$$\boldsymbol{\mathrm{so}}\::\: \\ $$$$\:\:\:\:\:\:\:\:\:{x}\:=\:\mathrm{11}−\sqrt{\frac{\mathrm{93}−\mathrm{13}\sqrt{\mathrm{17}}}{\mathrm{2}}}\:. \\ $$