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How to simplify the given equation and find the minimum value of $x$ ?
$$\frac{\sqrt{2x^2-1} + \sqrt{x^2-1}}{\sqrt2x^2}=1$$ I do square both sides but I doesn't make any sense

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  • $\begingroup$ $t=x^2$ makes stuff easier. $\endgroup$ – Trebor Apr 26 '19 at 6:22
  • $\begingroup$ And after that, use $\beta=2t^2-3t+2$ and obtain the result. $\endgroup$ – Trebor Apr 26 '19 at 6:25
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Set $t=x^2$ and clear denominators to get $$\sqrt{2t-1}+\sqrt{t-1}=\sqrt{2}t.$$ Squaring both sides shows that \begin{eqnarray*} 2t^2&=&(2t-1)+2\sqrt{(2t-1)(t-1)}+(t-1)\\ &=&3t-2+2\sqrt{2t^2-3t+1}. \end{eqnarray*} Isolating the square root and squaring again shows that $$(2t^2-3t+2)^2=4(2t^2-3t+1).$$ Rearranging yields the quartic equation $$4t^4-12t^3+9t^2=t^2(2t-3)^2=0.$$ Of course $t=0$ is impossible, so we see that $t=\tfrac32$ and so $x=\sqrt{\tfrac32}$. This shows that the minimum value for $x$ equals $\sqrt{\tfrac32}$.

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Put $u = x^2 - 1, v = x^2 \implies \sqrt{u+v} + \sqrt{u} = v\sqrt{2}\implies \sqrt{u+v} = v\sqrt{2} - \sqrt{u}\implies u+v = 2v^2-2\sqrt{2}v\sqrt{u}+u\implies 2v^2 - v - 2\sqrt{2}v\sqrt{u} = 0\implies v(2v-1-2\sqrt{2}\sqrt{u}) = 0\implies v = 0\implies x^2=0 \implies x = 0$ , or $2v= 1+2\sqrt{2}\sqrt{u}\implies 2(u+1) = 1 +2\sqrt{2}\sqrt{u}\implies 2u -2\sqrt{2}\sqrt{u} + 1 = 0\implies (\sqrt{2u} - 1)^2 = 0\implies \sqrt{2u} = 1\implies 2u = 1\implies 2(x^2-1) = 1\implies 2x^2 = 3\implies x = \pm \dfrac{\sqrt{6}}{2}$ . But $x = 0$ is not a solution,so the solutions are : $ x=\pm \dfrac{\sqrt{6}}{2}$.

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  • $\begingroup$ $\dfrac{\sqrt{3}}{\sqrt{2}} = \dfrac{\sqrt{6}}{2}$. $\endgroup$ – DeepSea Apr 26 '19 at 6:26
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Let $u=\sqrt{2x^2-1}\ge0,v=\sqrt{x^2-1}\ge0$ $\implies u^2-v^2=x^2$

and $ u+v=\sqrt2(u^2-v^2)$

If $u+v=0,\sqrt{2x^2-1}=\sqrt{x^2-1}=0$ which is untenable

$\implies u-v=\dfrac1{\sqrt2}\implies \sqrt2v=\sqrt2u-1$

and $u^2-2v^2=1\implies u^2-(\sqrt2u-1)^2=1$

$\implies 0=(u-\sqrt2)^2$

$u(\ne0)=?$

$x^2=\dfrac{u^2+1}2=?$

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  • $\begingroup$ How does this answer the question? $\endgroup$ – Servaes Apr 26 '19 at 6:45
  • $\begingroup$ @Servaes, Can you find $u$ from here? $\endgroup$ – lab bhattacharjee Apr 26 '19 at 6:46
  • $\begingroup$ I just have a hard time parsing what is going on. You use a lot of '$\implies$' and $\iff$' when I think you mean something like "It follows that...". $\endgroup$ – Servaes Apr 26 '19 at 6:53
  • $\begingroup$ @Servaes, $$\iff$$ implies & is implied by. $\endgroup$ – lab bhattacharjee Apr 26 '19 at 7:09
  • $\begingroup$ I understand that, but for example, you write that $$u^2-2v^2=1\iff u^2-(\sqrt2u-1)^2=1,$$ which really doesn't make any sense to me, and is an abuse of the '$\iff$' symbol also by your definition. I'm guessing you mean to say something like $$"\text{Also note that we have } u^2-2v^2=1\text{, and substituting the above then yields } u^2-(\sqrt{2}u-1)^2=1."$$ I made this guess while already knowing the answer, and even then I've had to check a few other guesses on where these identities might have come from (my third guess was correct). $\endgroup$ – Servaes Apr 26 '19 at 7:16

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