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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?
$$\begin{cases}{\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{xy}}}+\sqrt{\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{xy}}+\mathrm{9}}=\mathrm{10}}\\{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}=\mathrm{7}}\end{cases} \\ $$$$\boldsymbol{\mathrm{How}}\:\boldsymbol{\mathrm{many}}\:\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{roots}}\:\boldsymbol{\mathrm{of}}\: \\ $$$$\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{equtions}}\:\boldsymbol{\mathrm{system}}? \\ $$
Answered by mrW2 last updated on 05/Feb/18
(√(x^2 −4x(x−7)))+(√((x−7)^2 +2x(x−7)+9))=10 (√(28x−3x^2 ))+(√(3x^2 −28x+59))=10 t=28x−3x^2 (√t)+(√(59−t))=10 (√(59−t))=10−(√t) 59−t=100+t−20(√t) 2t−20(√t)+41=0 (√t)=((10±3(√2))/2) t=(((10±3(√2))/2))^2 =((59±10×3(√2))/2)=((59)/2)±15(√2) 3x^2 −28x+t=0 D=28^2 −4×3t=10(43±18(√2))>0 ⇒x=((28±(√(10(43±18(√2)))))/6) ⇒y=((−14±(√(10(43±18(√2)))))/6) there are totally 4 solutions for (x,y).
$$\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}\left({x}−\mathrm{7}\right)}+\sqrt{\left({x}−\mathrm{7}\right)^{\mathrm{2}} +\mathrm{2}{x}\left({x}−\mathrm{7}\right)+\mathrm{9}}=\mathrm{10} \\ $$$$\sqrt{\mathrm{28}{x}−\mathrm{3}{x}^{\mathrm{2}} }+\sqrt{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{28}{x}+\mathrm{59}}=\mathrm{10} \\ $$$${t}=\mathrm{28}{x}−\mathrm{3}{x}^{\mathrm{2}} \\ $$$$\sqrt{{t}}+\sqrt{\mathrm{59}−{t}}=\mathrm{10} \\ $$$$\sqrt{\mathrm{59}−{t}}=\mathrm{10}−\sqrt{{t}} \\ $$$$\mathrm{59}−{t}=\mathrm{100}+{t}−\mathrm{20}\sqrt{{t}} \\ $$$$\mathrm{2}{t}−\mathrm{20}\sqrt{{t}}+\mathrm{41}=\mathrm{0} \\ $$$$\sqrt{{t}}=\frac{\mathrm{10}\pm\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$${t}=\left(\frac{\mathrm{10}\pm\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}}\right)^{\mathrm{2}} =\frac{\mathrm{59}\pm\mathrm{10}×\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}}=\frac{\mathrm{59}}{\mathrm{2}}\pm\mathrm{15}\sqrt{\mathrm{2}} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} −\mathrm{28}{x}+{t}=\mathrm{0} \\ $$$${D}=\mathrm{28}^{\mathrm{2}} −\mathrm{4}×\mathrm{3}{t}=\mathrm{10}\left(\mathrm{43}\pm\mathrm{18}\sqrt{\mathrm{2}}\right)>\mathrm{0} \\ $$$$\Rightarrow{x}=\frac{\mathrm{28}\pm\sqrt{\mathrm{10}\left(\mathrm{43}\pm\mathrm{18}\sqrt{\mathrm{2}}\right)}}{\mathrm{6}} \\ $$$$\Rightarrow{y}=\frac{−\mathrm{14}\pm\sqrt{\mathrm{10}\left(\mathrm{43}\pm\mathrm{18}\sqrt{\mathrm{2}}\right)}}{\mathrm{6}} \\ $$$${there}\:{are}\:{totally}\:\mathrm{4}\:{solutions}\:{for}\:\left({x},{y}\right). \\ $$
Commented by NECx last updated on 05/Feb/18
please mrw2 how can one know the number of sokutions in an equation?
$${please}\:{mrw}\mathrm{2}\:{how}\:{can}\:{one}\:{know} \\ $$$${the}\:{number}\:{of}\:{sokutions}\:{in}\:{an} \\ $$$${equation}? \\ $$
Commented by mrW2 last updated on 05/Feb/18
I think there is no general method. Just try to find the solutions.
$${I}\:{think}\:{there}\:{is}\:{no}\:{general}\:{method}. \\ $$$${Just}\:{try}\:{to}\:{find}\:{the}\:{solutions}. \\ $$

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