Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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.

$${I}\:{think}\:{there}\:{is}\:{no}\:{general}\:{method}. \\ $$$${Just}\:{try}\:{to}\:{find}\:{the}\:{solutions}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com