<s>[INST] Solve the equation ${\sqrt{4x^2 + 5x+1}} - 2{\sqrt {x^2-x+1}} = 9x-3$ I tried factoring the expression inside the square root, but that does not seem to help. Squaring the equation makes it even more terrible.
Can anyone provide a hint about what should be done? 
 [/INST] Notice that
$$\left({\sqrt{4x^2 + 5x+1}} - 2{\sqrt {x^2-x+1}}\right) \left({\sqrt{4x^2 + 5x+1}} + 2{\sqrt {x^2-x+1}}\right) = 9x-3.$$
Thus, 
$$9x-3 =  \left(9x-3\right)\left({\sqrt{4x^2 + 5x+1}} + 2{\sqrt {x^2-x+1}}\right) \implies $$
$$9x-3 = 0 \implies x=1/3$$ or
$${\sqrt{4x^2 + 5x+1}} + 2{\sqrt {x^2-x+1}} = 1.$$
But the minimal value of $x^2-x+1$ is $3/4$, which implies that 
$$ 2{\sqrt {x^2-x+1}} \geq \sqrt{3} > 1.$$
Therefore $x =1/3$ is the only real solution.
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