<s>[INST] Root equation - What am I missing? There's a problem of which I know the solution but not the solving process:

$(\sqrt{x} + 7)(\sqrt{x} - 1) = \frac{105}{4}$

I'm convinced that up to:

$x + 6\sqrt{x} - 7 = \frac{105}{4}$

everything is correct.
But afterwards, I never seem to be able to get to the solution of $x = \frac{49}{4}$
How can this equation be solved?
 [/INST] Hint $\ $ Put $\:z = \sqrt{x},\:$ clear denominators and rearrange to obtain the quadratic equation$\:0 = 4 z^2 + 24 z - 133 = (2z-7)(2z+19),\:$ solvable by the quadratic formula, or the Rational Root Test or  AC method, etc.
Alternatively, purely arithmetically, if it has rational root $\:\sqrt{x} = a/b\:$ in lowest terms then
$$ (\sqrt{x} +7)(\sqrt{x}-1)\ =\ \frac{a+7b}b\ \frac{a-b}{b}\ =\ \frac{3\cdot 5\cdot 7}4 $$
Since $\:b\:$ is coprime to $\:a,\:$ also $\:b\:$ is coprime to $\:a+7b,\ a-b,\:$ so we deduce  $\:b = 2.\:$  Hence
$$  (a + 14)\:(a-2)\ =\ 3\cdot 5\cdot 7 $$
So we seek a factorization of $\:3\cdot 5\cdot 7\:$ whose factors differ by $\:16 = a+14-(a-2).\:$ Checking the few possible factorizations we find $\:21\cdot 5\:\Rightarrow\:a=7,\:$ and $\:-5\:(-21)\:\Rightarrow\:a = -19.\:$ Notice that this solution depends on uniqueness of factorization (as does RRT and the AC method).
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