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Question Number 200379 by cortano12 last updated on 18/Nov/23

Commented by Frix last updated on 18/Nov/23
$No exact solutions possible. Transform to { ((y^2 +((x^2 +x+1)/(2(x+1)))y+((x^2 +x−20)/(4(x+1)))=0)),((y^2 +((x^2 +6x+6)/(2x(x+1)))y+((3x−20)/(2x(x+1)))=0)) :} Subtract and solve for y y=−(((x−2)(x^2 +3x−20))/(2(x^3 −5x−6))) Insert and transform x^6 −35x^5 +34x^4 +500x^3 −784x^2 −680x+560=0 This has got 6 real solutions...$
$No exact solutions possible . Transform to$ ${\begin{cases} y^{2} + \frac{x^{2} + x + 1}{2 (x + 1)} y + \frac{x^{2} + x - 20}{4 (x + 1)} = 0 \\ y^{2} + \frac{x^{2} + 6 x + 6}{2 x (x + 1)} y + \frac{3 x - 20}{2 x (x + 1)} = 0 \end{cases}$ $Subtract and solve for y$ $y = - \frac{(x - 2) (x^{2} + 3 x - 20)}{2 (x^{3} - 5 x - 6)}$ $Insert and transform$ $x^{6} - 35 x^{5} + 34 x^{4} + 500 x^{3} - 784 x^{2} - 680 x + 560 = 0$ $This has got 6 real solutions . . .$
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