Solving radical and polynomial expressions

Question

$2x^3 +3x^2 +2x+1 = x(2x+3)(\sqrt{x^2 + \frac{1}{x}} )$

Only solution i could find is x = -1, the LHS can be expressed as $(x+1)(2x^2+x+1)$ and the LHS has a $\sqrt{\frac{x^3+1}{x}}$ which has a factor of $\sqrt {(x+1)}$ .

I know that this is not the only solution, what are the others? I always have a hard time solving these kinds of problems especially if they become systems, what tips/tricks can you utilize in solving problems like this one?

Answer 1

Hint: Squaring the whole equation and factorizing we get $$\left( 2\,x-1 \right) \left( x+1 \right) \left( 4\,{x}^{2}+4\,x-1 \right) =0$$

Answer 2

Squaring both sides and substracting yields $$ 0=8x^4 + 12x^3 - 2x^2 - 5x + 1=(4x^2 + 4x - 1)(2x - 1)(x + 1) $$ and this can be easily solved.

Answer 3

If you square both sides you get $$ 8x^4+12x^3-2x^2-5x+1 = 0 $$ If you search for rational solutions, you will find $x=-1$ and $x=\frac 12$, and so the equation will be equivalent to $$ (x+1)(2x-1)(4x^2+4x-1)=0 $$ So we get to real solutions, $x=-1,\frac 12$, and a pair of complex solutions.