x-2-y-2-x-y-18-xy-x-y-7-Solve-

Question Number 218019 by dscm last updated on 26/Mar/25

x^{2} + y^{2} + x + y = 18

x y + x + y = 7

S o l v e .

Answered by Ghisom last updated on 26/Mar/25

u = x + y \land v = x y

\Leftrightarrow

x = \frac{u}{2} - \frac{\sqrt{u^{2} - 4 v}}{2} \land y = \frac{u}{2} + \frac{\sqrt{u^{2} - 4 v}}{2}

(1) u^{2} + u - 2 v = 18

(2) u + v = 7 \Rightarrow v = 7 - u

insert in (1) & transform

u^{2} + 3 u - 32 = 0

u = - \frac{3}{2} \pm \frac{\sqrt{137}}{2}

v = \frac{17}{2} \mp \frac{\sqrt{137}}{2}

x = - \frac{3}{4} \pm \frac{\sqrt{137}}{4} - \frac{\sqrt{10 \pm \sqrt{137}}}{4}

y = - \frac{3}{4} \pm \frac{\sqrt{137}}{4} + \frac{\sqrt{10 \pm \sqrt{137}}}{4}

those with “ -^{″} are complex

those with “ +^{″} are real

(you can also exchange x ⇄ y)

Commented by MathematicalUser2357 last updated on 28/Mar/25

Here is some approximations :

Answered by Rasheed.Sindhi last updated on 26/Mar/25

x^{2} + y^{2} + x + y = 18 \land x y + x + y = 7

{(x + y)}^{2} - 2 x y + (x + y) = 18

∵ x y = 7 - (x + y)

∴ {(x + y)}^{2} - 2 (7 - (x + y)) + (x + y) = 18

∴ {(x + y)}^{2} - 14 + 2 (x + y)) + (x + y) = 18

∴ {(x + y)}^{2} + 3 (x + y)) - 32 = 0

x + y = \frac{- 3 \pm \sqrt{9 + 128}}{2}

= \frac{- 3 \pm \sqrt{137}}{2}

x y = 7 - (x + y) = 7 - \frac{- 3 \pm \sqrt{137}}{2}

= \frac{17 \mp \sqrt{137}}{2}

y = \frac{17 \mp \sqrt{137}}{2 x}

x = \frac{- 3 \pm \sqrt{137}}{2} - \frac{17 \mp \sqrt{137}}{2 x}

2 x^{2} - (- 3 \pm \sqrt{137}) x + 17 \mp \sqrt{137} = 0

{\begin{cases} 2 x^{2} - (- 3 + \sqrt{137}) x + 17 - \sqrt{137} = 0 \\ 2 x^{2} - (- 3 - \sqrt{137}) x + 17 + \sqrt{137} = 0 \end{cases}

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