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x-2-2y-2-xy-37-y-2-2x-2-2xy-26-find-x-2-y-2-




Question Number 207249 by hardmath last updated on 10/May/24
 { ((x^2   +  2y^2   +  xy  =  37)),((y^2   +  2x^2   +  2xy  =  26)) :}    find:  x^2  + y^2  = ?
$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{2y}^{\mathrm{2}} \:\:+\:\:\mathrm{xy}\:\:=\:\:\mathrm{37}}\\{\mathrm{y}^{\mathrm{2}} \:\:+\:\:\mathrm{2x}^{\mathrm{2}} \:\:+\:\:\mathrm{2xy}\:\:=\:\:\mathrm{26}}\end{cases}\:\:\:\:\mathrm{find}:\:\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:? \\ $$
Answered by A5T last updated on 10/May/24
(x^2 +y^2 +xy)=21...(i)  y^2 −x^2 −xy=11...(ii)  ⇒(i)+(ii): 2y^2 =32⇒y^2 =16⇒y=+_− 4  y=4⇒x^2 +_− 4x−5=0⇒x=−5 or 1 or 5 or 1  x^2 =25 or 1⇒x^2 +y^2 =41 or 17.
$$\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{xy}\right)=\mathrm{21}…\left({i}\right) \\ $$$${y}^{\mathrm{2}} −{x}^{\mathrm{2}} −{xy}=\mathrm{11}…\left({ii}\right) \\ $$$$\Rightarrow\left({i}\right)+\left({ii}\right):\:\mathrm{2}{y}^{\mathrm{2}} =\mathrm{32}\Rightarrow{y}^{\mathrm{2}} =\mathrm{16}\Rightarrow{y}=\underset{−} {+}\mathrm{4} \\ $$$${y}=\mathrm{4}\Rightarrow{x}^{\mathrm{2}} \underset{−} {+}\mathrm{4}{x}−\mathrm{5}=\mathrm{0}\Rightarrow{x}=−\mathrm{5}\:{or}\:\mathrm{1}\:{or}\:\mathrm{5}\:{or}\:\mathrm{1} \\ $$$${x}^{\mathrm{2}} =\mathrm{25}\:{or}\:\mathrm{1}\Rightarrow{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{41}\:{or}\:\mathrm{17}. \\ $$
Commented by hardmath last updated on 10/May/24
thank you dear professor cool
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{cool} \\ $$

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