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Question Number 193922 by sonukgindia last updated on 23/Jun/23

	Answered by talminator2856792 last updated on 23/Jun/23

	$x^{2} + y^{2} + 3 x y = 158$ $x^{2} (y^{2} + 1) + 2 x y + y^{2} = 1028$ $x^{2} + y^{2} + 3 x y + x^{2} y^{2} - x y = 1028$ $x^{2} y^{2} - x y + 158 = 1028$ $x^{2} y^{2} - x y = 870$ $x^{2} y^{2} - x y - 870 = 0$ ${(x y - \frac{1}{2})}^{2} - 870 - \frac{1}{4} = 0$ ${(x y - \frac{1}{2})}^{2} = \frac{3481}{4}$ $x y - \frac{1}{2} = \pm \frac{59}{2}$ $x y = \frac{60}{2}$ $x y = 30$
	Answered by Rajpurohith last updated on 23/Jun/23

	$G i v e n (x^{2} + y^{2} + 2 x y) + x y = 158 \to (1)$ $A n d x^{2} y^{2} + (x^{2} + 2 x y + y^{2}) = 1028 \to (2)$ $S u b t r a c t (1) f r o m (2)$ $\Rightarrow x^{2} y^{2} - x y = 1028 - 158 = 870$ $\Rightarrow t^{2} - t - 870 = 0 w h e r e t = x y .$ $t = \frac{1 \pm \sqrt{1 + 4 (870)}}{2} = \frac{1 \pm 59}{2} = 30, - 29$ $∙ x y = 30$ $f r o m (1) x^{2} + y^{2} = 158 - 3 (30) = 68$ $a n d w e h a v e x^{2} y^{2} = 900$ $s o a + b = 68 a n d a b = 900$ $\Rightarrow l^{2} - 68 l + 900 = 0$ $\Rightarrow l = \frac{68 \pm \sqrt{4624 - 4 (900)}}{2} = \frac{68 \pm \sqrt{1024}}{2} = \frac{68 \pm 32}{2}$ $= 50, 18 \Rightarrow a = 50 a n d b = 18$ $\Rightarrow x^{2} = 50 a n d y^{2} = 18$ $\Rightarrow x = \pm 5 \sqrt{2} a n d y = \pm 3 \sqrt{2} ◼$ $S i m i l a r l y y o u c a n w o r k f o r t h e c a s e x y = - 29 .$
	Answered by MM42 last updated on 23/Jun/23
	$x+y=s & xy=p { ((s^2 +p=158)),((p^2 +s^2 =1028)) :} ⇒p^2 −p−870=0⇒p=30 or −29 1)⇒p=30 →s^2 =128→s=8(√2)⇒X^2 −8(√2)X+30=0→... or s=−8(√2)⇒X^2 +8(√2)X+30=0→.. 2) p=−29 ⇒s^2 =187→s=(√(187))→X^2 −(√(187))X−29=0 or s=−(√(187))→X^2 +(√(187))X−29=0→...$
	$x + y = s & x y = p$ ${\begin{cases} s^{2} + p = 158 \\ p^{2} + s^{2} = 1028 \end{cases} \Rightarrow p^{2} - p - 870 = 0 \Rightarrow p = 30 o r - 29$ $1) \Rightarrow p = 30 \to s^{2} = 128 \to s = 8 \sqrt{2} \Rightarrow X^{2} - 8 \sqrt{2} X + 30 = 0 \to . . .$ $o r s = - 8 \sqrt{2} \Rightarrow X^{2} + 8 \sqrt{2} X + 30 = 0 \to . .$ $2) p = - 29 \Rightarrow s^{2} = 187 \to s = \sqrt{187} \to X^{2} - \sqrt{187} X - 29 = 0$ $o r s = - \sqrt{187} \to X^{2} + \sqrt{187} X - 29 = 0 \to . . .$
	Answered by Frix last updated on 23/Jun/23
	$Let x=p−q∧y=p+q { ((5p^2 −q^2 −158=0 ⇔ q^2 =5p^2 −158)),((p^4 −2p^2 (q^2 −2)+q^4 −1028=0)) :} p^4 −((315p^2 )/4)+1496=0 p=±((√(187))/2)∨p=±4(√2) ⇒ q=±((√(303))/2)∨q=±(√2) ⇒ x=±(((√(187))+(√(303)))/2)∧y=±(((√(187))−(√(303)))/2) Also exchanging x↔y solves both equations x=±3(√2)∧y=∓5(√2)$
	$Let x = p - q \land y = p + q$ ${\begin{cases} 5 p^{2} - q^{2} - 158 = 0 \Leftrightarrow q^{2} = 5 p^{2} - 158 \\ p^{4} - 2 p^{2} (q^{2} - 2) + q^{4} - 1028 = 0 \end{cases}$ $p^{4} - \frac{315 p^{2}}{4} + 1496 = 0$ $p = \pm \frac{\sqrt{187}}{2} \lor p = \pm 4 \sqrt{2}$ $\Rightarrow$ $q = \pm \frac{\sqrt{303}}{2} \lor q = \pm \sqrt{2}$ $\Rightarrow$ $x = \pm \frac{\sqrt{187} + \sqrt{303}}{2} \land y = \pm \frac{\sqrt{187} - \sqrt{303}}{2}$ $Also exchanging x \leftrightarrow y solves both equations$ $x = \pm 3 \sqrt{2} \land y = \mp 5 \sqrt{2}$
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