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1) $x^2+m^2=2mx+(nx)^2 $

2) $\sqrt{x^2+3x+7}-\sqrt{x^2-3x+9}+2=0$

I know the answer to the first equation is {$\frac{m}{1-n}, \frac{m}{1+n}$}. The answer to the second equation is {$-\frac{9}{5}$}.

I don't understand why I'm struggling on problems that look so simple. Here's how I approached the first problem:

$$x^2+m^2-2mx=n^2(x^2)$$

$$x^2+2mx+m^2-2mx=n^2x^2$$

$$\sqrt{x^2+m^2}=\sqrt{n^2x^2}$$

$$x=\frac{x+m}{n}, \frac{x-m}{n}$$

I suspect my answer is actually correct but it is only translated differently. I'm not too sure. I did plug my answer into a simplified version of the problem and indeed, I got, $x=\frac{x+m}{n}, \frac{x-m}{n}$ but I don't think that's trustworthy.

My approach to the second problem goes as follows:

$$\sqrt{x^2+3x+7}-\sqrt{x^2-3x+9}=-2$$

$$(\sqrt{x^2+3x+7})^2-(\sqrt{x^2-3x+9})^2=(-2)^2$$

$$x^2+3x+7-x^2+3x-9=4$$

$$6x=6$$

$$x=1$$

I don't believe squaring the roots actually works the way I tried here, but this was the best I could do. I tried factoring the polynomials within the roots individually, but that led to some really messy results so I avoided it.

Could someone please explain each step of either problem using TeX commands? Other people I asked either gave vague answers without much explanation or did part of the problem in hopes I would connect the dots and complete the rest. I'd really just appreciate a straight forward, detailed solve. Also, I would prefer if you do not skip steps - I don't mind if it's very long.