Question

Please help me with this question thank you

Answer 1

Answer:

$\mathrm{The\:solution\:is}$ :

$x=\frac{-y^2-1}{2-y^2}\space\left\{y\ge \:0\right\}$

Step-by-step explanation:

Given

$y=\sqrt{\frac{2x+1}{x-1}}$

Taking square of both sides

$y^2=\left(\sqrt{\frac{2x+1}{x-1}}\right)\:^2$

$\mathrm{Subtract\:}\left(\sqrt{\frac{2x+1}{x-1}}\right)^2\mathrm{\:from\:both\:sides}$

$y^2-\left(\sqrt{\frac{2x+1}{x-1}}\right)^2=\left(\sqrt{\frac{2x+1}{x-1}}\right)^2-\left(\sqrt{\frac{2x+1}{x-1}}\right)^2$

$\mathrm{Simplify}$

$y^2-\left(\sqrt{\frac{2x+1}{x-1}}\right)^2=0$

As we know that $\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}x^2-y^2=\left(x+y\right)\left(x-y\right)$

$\mathrm{Factor\:}y^2-\left(\sqrt{\frac{2x+1}{x-1}}\right)^2:\quad \left(y+\sqrt{\frac{2x+1}{x-1}}\right)\left(y-\sqrt{\frac{2x+1}{x-1}}\right)$

so

$\left(y+\sqrt{\frac{2x+1}{x-1}}\right)\left(y-\sqrt{\frac{2x+1}{x-1}}\right)=0$        

$\mathrm{Using\:the\:Zero\:Factor\:Principle:\quad \:If}\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)$

$\mathrm{Solve\:}\:y+\sqrt{\frac{2x+1}{x-1}}=0$

$\mathrm{Subtract\:}y\mathrm{\:from\:both\:sides}$

$y+\sqrt{\frac{2x+1}{x-1}}-y=0-y$

$\sqrt{\frac{2x+1}{x-1}}=-y$

$\mathrm{Square\:both\:sides}$

$\left(\sqrt{\frac{2x+1}{x-1}}\right)^2=\left(-y\right)^2$

$\mathrm{Expand\:}\left(\sqrt{\frac{2x+1}{x-1}}\right)^2$

$\left(\sqrt{\frac{2x+1}{x-1}}\right)^2$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt{a}=a^{\frac{1}{2}}$

$=\left(\left(\frac{2x+1}{x-1}\right)^{\frac{1}{2}}\right)^2$

$=\frac{2x+1}{x-1}$

so equation  $\left(\sqrt{\frac{2x+1}{x-1}}\right)^2=\left(-y\right)^2$ becomes

$\frac{2x+1}{x-1}=y^2$

now

$\mathrm{Solve\:}\:\frac{2x+1}{x-1}=y^2$

$\frac{2x+1}{x-1}=y^2$

$\mathrm{Multiply\:both\:sides\:by\:}x-1$

$\frac{2x+1}{x-1}\left(x-1\right)=y^2\left(x-1\right)$

$2x+1=y^2\left(x-1\right)$

$2x+1=xy^2-y^2$         ∵  $y^2\left(x-1\right):\quad xy^2-y^2$

$2x=xy^2-y^2-1$

$2x-xy^2=-y^2-1$

$x\left(2-y^2\right)=-y^2-1$         ∵ $\mathrm{Factor}\:2x-xy^2:\quad x\left(2-y^2\right)$

$\mathrm{Divide\:both\:sides\:by\:}2-y^2$

$\frac{x\left(2-y^2\right)}{2-y^2}=-\frac{y^2}{2-y^2}-\frac{1}{2-y^2}$

$x=\frac{-y^2-1}{2-y^2}$

so

$y+\sqrt{\frac{2x+1}{x-1}}=0:\quad x=\frac{-y^2-1}{2-y^2}\space\left\{y\le \:0\right\}$

similarly

$y-\sqrt{\frac{2x+1}{x-1}}=0:\quad x=\frac{-y^2-1}{2-y^2}\space\left\{y\ge \:0\right\}$

$\mathrm{Verify\:Solutions}:\quad x=\frac{-y^2-1}{2-y^2}$

$\mathrm{Check\:the\:solutions\:by\:plugging\:them\:into\:}y^2=\left(\sqrt{\frac{2x+1}{x-1}}\right)^2$

$\mathrm{Remove\:the\:ones\:that\:don't\:agree\:with\:the\:equation.}$

$\mathrm{Plug}\quad x=\frac{-y^2-1}{2-y^2}$

$y^2=\left(\sqrt{\frac{2\left(\frac{-y^2-1}{2-y^2}\right)+1}{\left(\frac{-y^2-1}{2-y^2}\right)-1}}\right)^2$

$\mathrm{Subtract\:}\left(\sqrt{\frac{2\frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)^2\mathrm{\:from\:both\:sides}$

$y^2-\left(\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)^2=\left(\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)^2-\left(\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)^2$

$y^2-\left(\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)^2=0$

$\mathrm{Factor\:}y^2-\left(\sqrt{\frac{2\frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)^2:\quad \left(y+\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)\left(y-\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)$

so

$\left(y+\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)\left(y-\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}\right)=0$

$\mathrm{Using\:the\:Zero\:Factor\:Principle:\quad \:If}\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)$

$\mathrm{Solve\:}\:y+\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}=0:\quad y\le \:0$

$\mathrm{Solve\:}\:y-\sqrt{\frac{2\cdot \frac{-y^2-1}{2-y^2}+1}{\frac{-y^2-1}{2-y^2}-1}}=0:\quad y\ge \:0$

$\mathrm{True\:for\:all}\:y$

Therefore,  $\mathrm{The\:solution\:is}$ :

$x=\frac{-y^2-1}{2-y^2}\space\left\{y\ge \:0\right\}$