{ (((√(x^2 −4xy))+(√(y^2 +2xy+9))=10)),((x−y=7)) :} How many real roots of the equtions system?


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Question Number 29184 by ANTARESVY last updated on 05/Feb/18
${ (((√(x^2 −4xy))+(√(y^2 +2xy+9))=10)),((x−y=7)) :} How many real roots of the equtions system?$
${\begin{cases} \sqrt{x^{2} - 4 xy} + \sqrt{y^{2} + 2 xy + 9} = 10 \\ x - y = 7 \end{cases}$ $How many real roots of$ $the equtions system ?$
	Answered by mrW2 last updated on 05/Feb/18

	$\sqrt{x^{2} - 4 x (x - 7)} + \sqrt{{(x - 7)}^{2} + 2 x (x - 7) + 9} = 10$ $\sqrt{28 x - 3 x^{2}} + \sqrt{3 x^{2} - 28 x + 59} = 10$ $t = 28 x - 3 x^{2}$ $\sqrt{t} + \sqrt{59 - t} = 10$ $\sqrt{59 - t} = 10 - \sqrt{t}$ $59 - t = 100 + t - 20 \sqrt{t}$ $2 t - 20 \sqrt{t} + 41 = 0$ $\sqrt{t} = \frac{10 \pm 3 \sqrt{2}}{2}$ $t = {(\frac{10 \pm 3 \sqrt{2}}{2})}^{2} = \frac{59 \pm 10 \times 3 \sqrt{2}}{2} = \frac{59}{2} \pm 15 \sqrt{2}$ $3 x^{2} - 28 x + t = 0$ $D = 28^{2} - 4 \times 3 t = 10 (43 \pm 18 \sqrt{2}) > 0$ $\Rightarrow x = \frac{28 \pm \sqrt{10 (43 \pm 18 \sqrt{2})}}{6}$ $\Rightarrow y = \frac{- 14 \pm \sqrt{10 (43 \pm 18 \sqrt{2})}}{6}$ $t h e r e a r e t o t a l l y 4 s o l u t i o n s f o r (x, y) .$
	Commented by NECx last updated on 05/Feb/18

	$p l e a s e m r w 2 h o w c a n o n e k n o w$ $t h e n u m b e r o f s o k u t i o n s i n a n$ $e q u a t i o n ?$
	Commented by mrW2 last updated on 05/Feb/18

	$I t h i n k t h e r e i s n o g e n e r a l m e t h o d .$ $J u s t t r y t o f i n d t h e s o l u t i o n s .$
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