x^2 + y^2 + xy + 2(x − y) = 9 How many solution that fulfilled the equation above ? x, y ∈ N


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Question Number 10755 by Joel576 last updated on 24/Feb/17

$x^{2} + y^{2} + x y + 2 (x - y) = 9$ $How many solution that fulfilled the equation above ?$ $x, y \in N$
	Answered by mrW1 last updated on 24/Feb/17
	$x^2 + y^2 − 2xy + 2(x − y)+3xy = 9 (x−y)^2 +2(x−y)+3xy−9=0 u=x−y v=xy ⇒u^2 +2u+3v−9=0 u=((−2±(√(4−4(3v−9))))/2)=−1±(√(10−3v)) 10−3v=k^2 v=((10−k^2 )/3)>0 k=1: v=((10−1^2 )/3)=3 ok k=2: v=((10−2^2 )/3)=2 ok k=3: v=((10−3^2 )/3)=(1/3) not ok with v=3 u=−1±1= { (0),((−2)) :} { ((x−y=0),( )),((xy=3),( ⇒no integer solution!)) :} { ((x−y=−2),(⇒x=1, y=3)),((xy=3),(⇒ right solution!)) :} with v=2 u=−1±2= { (1),((−3)) :} { ((x−y=1),(⇒x=2, y=1)),((xy=2),(⇒ right solution!)) :} { ((x−y=−3),),((xy=2),(⇒ no integer solution!)) :} all solutions: ((x),(y) )= ((1),(3) ) or ((2),(1) )$
	$x^{2} + y^{2} - 2 x y + 2 (x - y) + 3 x y = 9$ ${(x - y)}^{2} + 2 (x - y) + 3 x y - 9 = 0$ $u = x - y$ $v = x y$ $\Rightarrow u^{2} + 2 u + 3 v - 9 = 0$ $u = \frac{- 2 \pm \sqrt{4 - 4 (3 v - 9)}}{2} = - 1 \pm \sqrt{10 - 3 v}$ $10 - 3 v = k^{2}$ $v = \frac{10 - k^{2}}{3} > 0$ $k = 1 : v = \frac{10 - 1^{2}}{3} = 3 o k$ $k = 2 : v = \frac{10 - 2^{2}}{3} = 2 o k$ $k = 3 : v = \frac{10 - 3^{2}}{3} = \frac{1}{3} n o t o k$ $w i t h v = 3$ $u = - 1 \pm 1 = {\begin{cases} 0 \\ - 2 \end{cases}$ ${\begin{cases} x - y = 0 \\ x y = 3 & \Rightarrow n o i n t e g e r s o l u t i o n! \end{cases}$ ${\begin{cases} x - y = - 2 & \Rightarrow x = 1, y = 3 \\ x y = 3 & \Rightarrow r i g h t s o l u t i o n! \end{cases}$ $w i t h v = 2$ $u = - 1 \pm 2 = {\begin{cases} 1 \\ - 3 \end{cases}$ ${\begin{cases} x - y = 1 & \Rightarrow x = 2, y = 1 \\ x y = 2 & \Rightarrow r i g h t s o l u t i o n! \end{cases}$ ${\begin{cases} x - y = - 3 \\ x y = 2 & \Rightarrow n o i n t e g e r s o l u t i o n! \end{cases}$ $a l l s o l u t i o n s :$ $(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 1 \\ 3 \end{matrix}) o r (\begin{matrix} 2 \\ 1 \end{matrix})$
	Commented by Joel576 last updated on 24/Feb/17

	$t h a n k y o u v e r y m u c h$
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