interesting equation

[Guest]

How to solve this?
[tex]\sqrt{2x^{2}-4x+23}+\sqrt{5x^{2}-10x+9}=-x^{2}+2x+8[/tex]

[Guest]

I would start by getting rid of the square roots by squaring! Of course, since [tex](a+ b)^2= a^2+ ab+ b^2[/tex] just squaring both sides would not get rid of the square roots- they would appear in the "2ab" term.

Instead subtract one of the square roots from both sides to get one on each side, then square:
[tex]\sqrt{2x^2- 4x+ 23}= -x^2+ 2x+ 8- \sqrt{5x^2- 10x+ 9}[/tex]
Squaring
[tex]2x^2- 4x+ 23= (-x^2+ 2x+ 8)^2+ 2(x^2- 2x- 8)\sqrt{5x^2- 10x+ 9}+ 5x^2- 10x+ 9[/tex]
[tex]2x^2- 4x+ 23= x^4+ 4x^2+ 64- 4x^3- 16x^2+ 32x+ 5x^2- 10x+ 9+ (2x^2- 4x- 16)\sqrt{x^2- 2x+ 8}[/tex]
Get everything except the remaining square root on one side, the square root on the other, and square both side. That's going to get very complicated, resulting in an eight degree polynomial- and it's not my problem so I'm going to stop here!