Answer
$x=6,-1$
Work Step by Step
To be a right triangle, one set of slopes must be opposite reciprocals, for this would make two of the lines perpendicular. Thus:
$m_1 = \frac{y_2-y_1}{x_2-x_1} = \frac{x-2}{4-2} = (x-2)/4$
$m_2 = \frac{y_2-y_1}{x_2-x_1} = \frac{x-3}{4-7} =(x-3)/(-3) $
We set the opposite reciprocals equal to each other:
$ \frac{-4}{x-2} = \frac{x-3}{-3} \\ 12 = x^2 -5x +6 \\ 0 = x^2 -5x -6 \\ 0= (x-6)(x+1)\\ x= 6, -1$