Square root equation problem $2\sqrt{2x}-\sqrt{x-1}-\sqrt{x+7}=0$

Question

I was solving some equations and got to this one:

$$2\sqrt{2x}-\sqrt{x-1}-\sqrt{x+7}=0$$

I tried square the equation, but it seems that it only complicates the process.

$x=2$ works... Worth noticing the easy solutions first, helps make the process concrete — Will Jagy, Sep 08 '18 at 18:28

score 5 · Accepted Answer · answered Sep 08 '18 at 17:04

5

Hint: Writing your equation in the form

$$2\sqrt{2x}=\sqrt{x-1}+\sqrt{x+7}$$ and squaring gives

$$8x=x-1+x+7+2\sqrt{x-1}\sqrt{x+7}$$ Can you finish? Simplifying and squaring once more we get $$9(x-1)^2=(x-1)(x+7)$$

answered Sep 08 '18 at 17:04

1

worth noting, for the case $x > 1,$ your final equation becomes 9 = Mobius transformation(x), and inverting Mobius transformations is a surprisingly easy technique, worth learning even for beginners – Will Jagy Sep 08 '18 at 18:33
Do you know any example of inverting with mobius transformation, it seems that I can't find anything like that – VLC Sep 09 '18 at 12:03

1 Answers1