Answer
$x= 2,6$
Work Step by Step
$\sqrt {x-2} +3=\sqrt {4x+1}$
$\sqrt {x-2} +3-3=\sqrt {4x+1}-3$
$\sqrt {x-2} =\sqrt {4x+1}-3$
$(\sqrt {x-2})^2 =(\sqrt {4x+1}-3)^2$
$x-2=(\sqrt {4x+1})*(\sqrt {4x+1})+(\sqrt {4x+1})*(-3)+(-3)(\sqrt {4x+1})+(-3)(-3)$
$x-2=(4x+1)-6\sqrt {4x+1}+9$
$x-2=4x+10-6\sqrt {4x+1}$
$-3x-2=10-6\sqrt {4x+1}$
$-3x-12=-6\sqrt {4x+1}$
$(-3x-12)^2=(-6\sqrt {4x+1})^2$
$(-3x)^2+(-3x)(-12)+(-12)(-3x)+(-12)(-12)=(-6)^2(\sqrt {4x+1})^2$
$9x^2+72x+144=36(4x+1)$
$9x^2+72x+144=144x+36$
$9x^2-72x+144=36$
$9x^2-72x+108=0$
$(9x^2-72x+108=0)/9$
$x^2-8x+12=0$
$(x-2)(x-6)=0$
$x-2=0$
$x=2$
$x-6=0$
$x=6$
$\sqrt {x-2} +3=\sqrt {4x+1}$
$\sqrt {2-2} +3=\sqrt {4*2+1}$
$\sqrt {0} +3=\sqrt {8+1}$
$0+3=\sqrt {9}$
$0+3=3$
$3=3$ (true)
$\sqrt {x-2} +3=\sqrt {4x+1}$
$\sqrt {6-2} +3=\sqrt {4*6+1}$
$\sqrt {4} +3=\sqrt {24+1}$
$2 +3=\sqrt {25}$
$5=5$ (true)
