x(y+1) + y(x−1) = 12 x(√y) + y(√x) = 21 x, y ∈ R x + y = ?


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Question Number 61565 by naka3546 last updated on 04/Jun/19

$x (y + 1) + y (x - 1) = 12$ $x \sqrt{y} + y \sqrt{x} = 21$ $x, y \in R$ $x + y = ?$
Commented by MJS last updated on 05/Jun/19

${what}^{'} s the source of this question ?$ ${it}^{'} s got no exact solution . . .$
Commented by naka3546 last updated on 05/Jun/19

$I {d o n}^{'} t k n o w, s i r .$ $\frac{12}{x y} = \frac{21}{\sqrt{x y}} - 2 \sqrt{x y}$ $x + y = ?$
Commented by MJS last updated on 05/Jun/19

$this is only one equation in 2 variables but$ $I {don}^{'} t think you can get it from the first two . . .$ $\sqrt{x y} = t [t ⩾ 0]$ $\frac{12}{t^{2}} = \frac{21}{t} - 2 t$ $t^{3} - \frac{21}{2} t + 6 = 0$ $t_{0} = \sqrt{14} \sin (\frac{1}{3} \arcsin \frac{6 \sqrt{14}}{49}) \approx . 591098$ $t_{1} = \sqrt{14} \cos (\frac{π}{6} + \frac{1}{3} \arcsin \frac{6 \sqrt{14}}{49}) \approx 2.90413$ $t_{2} = - \sqrt{14} \sin (\frac{π}{3} + \frac{1}{3} \arcsin \frac{6 \sqrt{14}}{49}) \approx - 3.49523 [not valid]$ $\Rightarrow x y = . 349397 \lor x y = 8.43398$
	Answered by MJS last updated on 05/Jun/19

	$x > 0 \land y > 0$ $(1) 2 x y + x - y - 12 = 0 \Rightarrow x = \frac{y + 12}{2 y + 1}$ $let x = u^{2} \land y = v^{2} \land u > 0 \land v > 0$ $(1) \Rightarrow u = \frac{\sqrt{v^{2} + 12}}{\sqrt{2 v^{2} + 1}}$ $(2) \Rightarrow \frac{v^{2} \sqrt{v^{2} + 12}}{\sqrt{2 v^{2} + 1}} + \frac{v (v^{2} + 12)}{2 v^{2} + 1} = 21$ $\frac{v (v^{2} + 12) + v^{2} \sqrt{(v^{2} + 12) (2 v^{2} + 1)}}{2 v^{2} + 1} = 21$ $v^{2} \sqrt{(v^{2} + 12) (2 v^{2} + 1)} = - v^{3} + 42 v^{2} - 12 v + 21$ $v^{4} (2 v^{4} + 25 v^{2} + 12) = {(v^{3} - 42 v^{2} + 12 v - 21)}^{2}$ $\Rightarrow v^{8} + 12 v^{6} + 42 v^{5} - 888 v^{4} + 525 v^{3} - 954 v^{2} + 252 v - \frac{441}{2} = 0$ $this has got only one root > 0$ $v = 4.448447 . . .$ $\Rightarrow y = 19.78864 . . .$ $\Rightarrow x = . 7834092 . . .$ $x + y = 20.57209 . . .$
	Answered by behi83417@gmail.com last updated on 05/Jun/19
	$x=p^2 ,y=q^2 ⇒ { ((p^2 −q^2 +2p^2 q^2 =12)),((pq(p+q)=21)) :} p+q=s,pq=t⇒ { ((s(√(s^2 −4t))+2t^2 =12)),((st=21)) :} s^2 (s^2 −4t)=144−48t^2 +4t^4 s^4 −84s=144−48×((441)/s^2 )+4×((194481)/s^4 ) s^8 −84s^5 −144s^4 +21168s^2 −777924=0 s=−4.631,5.334⇒t=−.4.535,3.937 ⇒ { ((p+q=−4.631)),((pq=−4.535)) :}⇒z^2 −4.631z−4.535=0 ⇒ { ((p=5.461⇒x=29.823)),((q=−0.83⇒y=0.689)) :}⇒x+y=30.512 ⇒ { ((p+q=5.334)),((pq=3.937⇒z^2 +5.334z+3.937=0)) :} ⇒ { ((p=−4.46⇒x=19.892)),((q=−0.884⇒y=0.778)) :}⇒x+y=20.67$
	$x = p^{2}, y = q^{2} \Rightarrow {\begin{cases} p^{2} - q^{2} + {2 p}^{2} q^{2} = 12 \\ pq (p + q) = 21 \end{cases}$ $p + q = s, pq = t \Rightarrow {\begin{cases} s \sqrt{s^{2} - 4 t} + {2 t}^{2} = 12 \\ st = 21 \end{cases}$ $s^{2} (s^{2} - 4 t) = 144 - {48 t}^{2} + {4 t}^{4}$ $s^{4} - 84 s = 144 - 48 \times \frac{441}{s^{2}} + 4 \times \frac{194481}{s^{4}}$ $s^{8} - {84 s}^{5} - {144 s}^{4} + {21168 s}^{2} - 777924 = 0$ $s = - 4.631, 5.334 \Rightarrow t = - . 4.535, 3.937$ $\Rightarrow {\begin{cases} p + q = - 4.631 \\ pq = - 4.535 \end{cases} \Rightarrow z^{2} - 4 . 631 z - 4.535 = 0$ $\Rightarrow {\begin{cases} p = 5.461 \Rightarrow x = 29.823 \\ q = - 0.83 \Rightarrow y = 0.689 \end{cases} \Rightarrow x + y = 30.512$ $\Rightarrow {\begin{cases} p + q = 5.334 \\ pq = 3.937 \Rightarrow z^{2} + 5 . 334 z + 3.937 = 0 \end{cases}$ $\Rightarrow {\begin{cases} p = - 4.46 \Rightarrow x = 19.892 \\ q = - 0.884 \Rightarrow y = 0.778 \end{cases} \Rightarrow x + y = 20.67$
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