Question Number 218019 by dscm last updated on 26/Mar/25
$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{x}+{y}=\mathrm{18} \\ $$$${xy}+{x}+{y}=\mathrm{7} \\ $$$${Solve}. \\ $$
Answered by Ghisom last updated on 26/Mar/25
$${u}={x}+{y}\wedge{v}={xy} \\ $$$$\Leftrightarrow \\ $$$${x}=\frac{{u}}{\mathrm{2}}−\frac{\sqrt{{u}^{\mathrm{2}} −\mathrm{4}{v}}}{\mathrm{2}}\wedge{y}=\frac{{u}}{\mathrm{2}}+\frac{\sqrt{{u}^{\mathrm{2}} −\mathrm{4}{v}}}{\mathrm{2}} \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:{u}^{\mathrm{2}} +{u}−\mathrm{2}{v}=\mathrm{18} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:{u}+{v}=\mathrm{7}\:\Rightarrow\:{v}=\mathrm{7}−{u} \\ $$$$\mathrm{insert}\:\mathrm{in}\:\left(\mathrm{1}\right)\:\&\:\mathrm{transform} \\ $$$${u}^{\mathrm{2}} +\mathrm{3}{u}−\mathrm{32}=\mathrm{0} \\ $$$${u}=−\frac{\mathrm{3}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{137}}}{\mathrm{2}} \\ $$$${v}=\frac{\mathrm{17}}{\mathrm{2}}\mp\frac{\sqrt{\mathrm{137}}}{\mathrm{2}} \\ $$$${x}=−\frac{\mathrm{3}}{\mathrm{4}}\pm\frac{\sqrt{\mathrm{137}}}{\mathrm{4}}−\frac{\sqrt{\mathrm{10}\pm\sqrt{\mathrm{137}}}}{\mathrm{4}} \\ $$$${y}=−\frac{\mathrm{3}}{\mathrm{4}}\pm\frac{\sqrt{\mathrm{137}}}{\mathrm{4}}+\frac{\sqrt{\mathrm{10}\pm\sqrt{\mathrm{137}}}}{\mathrm{4}} \\ $$$$\mathrm{those}\:\mathrm{with}\:“−''\:\mathrm{are}\:\mathrm{complex} \\ $$$$\mathrm{those}\:\mathrm{with}\:“+''\:\mathrm{are}\:\mathrm{real} \\ $$$$\left(\mathrm{you}\:\mathrm{can}\:\mathrm{also}\:\mathrm{exchange}\:{x}\:\rightleftarrows\:{y}\right) \\ $$
Commented by MathematicalUser2357 last updated on 28/Mar/25
$$\mathrm{Here}\:\mathrm{is}\:\mathrm{some}\:\mathrm{approximations}: \\ $$$$\: \\ $$
Answered by Rasheed.Sindhi last updated on 26/Mar/25
$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{x}+{y}=\mathrm{18}\:\wedge\:{xy}+{x}+{y}=\mathrm{7} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{2}{xy}+\left({x}+{y}\right)=\mathrm{18} \\ $$$$\because\:{xy}=\mathrm{7}−\left({x}+{y}\right) \\ $$$$\therefore\:\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{2}\left(\:\mathrm{7}−\left({x}+{y}\right)\:\right)+\left({x}+{y}\right)=\mathrm{18} \\ $$$$\left.\therefore\:\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{14}+\mathrm{2}\left({x}+{y}\right)\:\right)+\left({x}+{y}\right)=\mathrm{18} \\ $$$$\left.\therefore\:\left({x}+{y}\right)^{\mathrm{2}} +\mathrm{3}\left({x}+{y}\right)\:\right)−\mathrm{32}=\mathrm{0} \\ $$$${x}+{y}=\frac{−\mathrm{3}\pm\sqrt{\mathrm{9}+\mathrm{128}}}{\mathrm{2}} \\ $$$$\:\:\:\:\:=\frac{−\mathrm{3}\pm\sqrt{\mathrm{137}}}{\mathrm{2}} \\ $$$${xy}=\mathrm{7}−\left({x}+{y}\right)=\mathrm{7}−\frac{−\mathrm{3}\pm\sqrt{\mathrm{137}}}{\mathrm{2}} \\ $$$$\:\:\:\:=\frac{\mathrm{17}\mp\sqrt{\mathrm{137}}}{\mathrm{2}} \\ $$$${y}=\frac{\mathrm{17}\mp\sqrt{\mathrm{137}}}{\mathrm{2}{x}} \\ $$$${x}\:=\frac{−\mathrm{3}\pm\sqrt{\mathrm{137}}}{\mathrm{2}}−\frac{\mathrm{17}\mp\sqrt{\mathrm{137}}}{\mathrm{2}{x}} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} −\left(−\mathrm{3}\pm\sqrt{\mathrm{137}}\right){x}+\mathrm{17}\mp\sqrt{\mathrm{137}}\:=\mathrm{0} \\ $$$$\begin{cases}{\mathrm{2}{x}^{\mathrm{2}} −\left(−\mathrm{3}+\sqrt{\mathrm{137}}\right){x}+\mathrm{17}−\sqrt{\mathrm{137}}\:=\mathrm{0}}\\{\mathrm{2}{x}^{\mathrm{2}} −\left(−\mathrm{3}−\sqrt{\mathrm{137}}\right){x}+\mathrm{17}+\sqrt{\mathrm{137}}\:=\mathrm{0}}\end{cases} \\ $$$$… \\ $$