Answer
$x=6$, $x=2$
Work Step by Step
$\sqrt {x-2} = \sqrt {4x+1} -3$
$(\sqrt {x-2})^2 = (\sqrt {4x+1} -3)^2$
$x-2 = (\sqrt {4x+1}-3)*(\sqrt {4x+1}-3)$
$x-2 = \sqrt {4x+1}*\sqrt {4x+1}+(-3)(\sqrt {4x+1})+(-3)(\sqrt {4x+1})+(-3)*(-3)$
$x-2 = (4x+1) -6\sqrt {4x+1} + 9$
$x-2=4x+1+9-6\sqrt {4x+1}$
$x-2=4x+10-6\sqrt {4x+1}$
$x-2-4x-10=4x+10-6\sqrt {4x+1}-4x-10$
$-3x-12=-6\sqrt {4x+1}$
$(-3x-12)^2=(-6\sqrt {4x+1})^2$
$(-3x-12)*(-3x-12)=36*(4x+1)$
$(-3x)(-3x)+(-3x)(-12)+(-3x)(-12)+(-12)(-12) =36*4x+36$
$9x^2+36x+36x+144=144x+36$
$9x^2+72x+144=144x+36$
$9x^2+72x+144-144x-36=144x+36-144x-36$
$9x^2-72x+108=0$
$(9x^2-72x+108)/9=0/9$
$x^2-8x+12=0$
$(x-6)(x-2)=0$
$x-6=0$
$x-6+6=0+6$
$x=6$
$x-2=0$
$x-2+2=0+2$
$x=2$