Hello All of You verry Nice Day, God bless You love peace and happiness Solve for (x,y)∈R^2 { ((x^2 +y^2 =2x+3y+1)),((x^4 +y^4 =4x^2 +9y^2 +12xy+2x^2 y^2 +18)) :}


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Question Number 80448 by mind is power last updated on 03/Feb/20
$Hello All of You verry Nice Day, God bless You love peace and happiness Solve for (x,y)∈R^2 { ((x^2 +y^2 =2x+3y+1)),((x^4 +y^4 =4x^2 +9y^2 +12xy+2x^2 y^2 +18)) :}$
$H e l l o A l l o f Y o u v e r r y N i c e D a y, G o d b l e s s Y o u l o v e p e a c e a n d$ $h a p p i n e s s$ $S o l v e f o r (x, y) \in R^{2}$ ${\begin{cases} x^{2} + y^{2} = 2 x + 3 y + 1 \\ x^{4} + y^{4} = 4 x^{2} + 9 y^{2} + 12 x y + 2 x^{2} y^{2} + 18 \end{cases}$
Commented by mr W last updated on 03/Feb/20

$t h a n k y o u s i r! t h e s a m e t o y o u!$
Commented by john santu last updated on 03/Feb/20

$x^{4} + y^{4} = {(x^{2} + y^{2})}^{2} - 2 x^{2} y^{2}$ ${(2 x + 3 y + 1)}^{2} - 2 x^{2} y^{2} = 4 x^{2} + 9 y^{2} + 12 x y + 2 x^{2} y^{2} + 18$ $4 x^{2} + 9 y^{2} + 1 + 2 (6 x y + 2 x + 3 y) - 2 x^{2} y^{2} =$ $4 x^{2} + 9 y^{2} + 12 x y + 2 x^{2} y^{2} + 18$ $1 + 4 x + 6 y - 2 x^{2} y^{2} = 2 x^{2} y^{2} + 18$ $4 x^{2} y^{2} - 4 x - 6 y + 17 = 0$ $4 x^{2} y^{2} - 2 (2 x + 3 y) + 17 = 0$ $4 x^{2} y^{2} - 2 (x^{2} + y^{2} - 1) + 17 = 0$ $2 (x^{2} + y^{2}) - 4 x^{2} y^{2} - 19 = 0$ $l e t x^{2} = t, y^{2} = p$ $2 (t + p) - 4 t p - 19 = 0$ $n e x t . . .$
Commented by jagoll last updated on 03/Feb/20

$t o b e c o n t i n u e$
	Answered by MJS last updated on 03/Feb/20
	$let x=u−v∧y=u+v { ((2u^2 +2v^2 =5u+v+1)),((2u^4 +12u^2 v^2 +2v^4 =2u^4 −4u^2 v^2 +2v^4 +25u^2 +10uv+v^2 +18)) :} { ((2u^2 +2v^2 −5u−v−1=0)),((16u^2 v^2 −25u^2 −10uv−v^2 −18=0)) :} { ((v^2 −(v/2)+((2u^2 −5u−1)/2)=0)),((v^2 −((10u)/(16u^2 −1))v−((25u^2 +18)/(16u^2 −1))=0)) :} subtracting (1) − (2) and solving for v leads to v=((32u^4 −80u^3 +32u^2 +5u+37)/(16u^2 −20u−1)) inserting in (1) or (2) leads to a polynome of 8^(th) degree for u I can only solve approximately ⇒ (all rounded to 6 significant digits) x_1 =−.184612 y_1 =3.18722 x_2 =.323632 y_2 =3.44744 x_(3, 4) =−1.37362±.450762i y_(3, 4) =.705266∓1.34628i x_(5, 6) =.618559±1.80074 y_(5, 6) =−1.22229∓.252316i x_(7, 8) =2.68555±.324293i y_(7, 8) =.199692±.420370i these solutions give the folliwing polynomes: x^8 −4x^7 +2x^6 +6x^5 −16x^4 +15x^3 +((109)/2)x^2 −((17)/2)x−((53)/(16))=0 y^8 −6y^7 +7y^6 +9y^5 −((37)/2)y^4 +((45)/2)y^3 +((123)/4)y^2 −((51)/4)y+((137)/(16))=0$
	$let x = u - v \land y = u + v$ ${\begin{cases} 2 u^{2} + 2 v^{2} = 5 u + v + 1 \\ 2 u^{4} + 12 u^{2} v^{2} + 2 v^{4} = 2 u^{4} - 4 u^{2} v^{2} + 2 v^{4} + 25 u^{2} + 10 u v + v^{2} + 18 \end{cases}$ ${\begin{cases} 2 u^{2} + 2 v^{2} - 5 u - v - 1 = 0 \\ 16 u^{2} v^{2} - 25 u^{2} - 10 u v - v^{2} - 18 = 0 \end{cases}$ ${\begin{cases} v^{2} - \frac{v}{2} + \frac{2 u^{2} - 5 u - 1}{2} = 0 \\ v^{2} - \frac{10 u}{16 u^{2} - 1} v - \frac{25 u^{2} + 18}{16 u^{2} - 1} = 0 \end{cases}$ $subtracting (1) - (2) and solving for v leads to$ $v = \frac{32 u^{4} - 80 u^{3} + 32 u^{2} + 5 u + 37}{16 u^{2} - 20 u - 1}$ $inserting in (1) or (2) leads to a polynome of$ $8^{th} degree for u I can only solve approximately$ $\Rightarrow$ $(all rounded to 6 significant digits)$ $x_{1} = - . 184612 y_{1} = 3.18722$ $x_{2} = . 323632 y_{2} = 3.44744$ $x_{3, 4} = - 1.37362 \pm . 450762 i y_{3, 4} = . 705266 \mp 1 . 34628 i$ $x_{5, 6} = . 618559 \pm 1.80074 y_{5, 6} = - 1.22229 \mp . 252316 i$ $x_{7, 8} = 2.68555 \pm . 324293 i y_{7, 8} = . 199692 \pm . 420370 i$ $these solutions give the folliwing polynomes :$ $x^{8} - 4 x^{7} + 2 x^{6} + 6 x^{5} - 16 x^{4} + 15 x^{3} + \frac{109}{2} x^{2} - \frac{17}{2} x - \frac{53}{16} = 0$ $y^{8} - 6 y^{7} + 7 y^{6} + 9 y^{5} - \frac{37}{2} y^{4} + \frac{45}{2} y^{3} + \frac{123}{4} y^{2} - \frac{51}{4} y + \frac{137}{16} = 0$
	Commented by mind is power last updated on 04/Feb/20

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