Answer
The solutions is $x=\dfrac{13}{6}+\dfrac{\sqrt{105}}{6}$
Work Step by Step
$\sqrt{3\sqrt{x+1}}=\sqrt{3x-5}$
Square both sides of the equation:
$(\sqrt{3\sqrt{x+1}})^{2}=(\sqrt{3x-5})^{2}$
$3\sqrt{x+1}=3x-5$
Square both sides of the equation again:
$(3\sqrt{x+1})^{2}=(3x-5)^{2}$
$9(x+1)=9x^{2}-30x+25$
$9x+9=9x^{2}-30x+25$
Take all terms to the right side:
$0=9x^{2}-30x+25-9x-9$
Rearrange and simplify:
$9x^{2}-30x+25-9x-9=0$
$9x^{2}-39x+16=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
In this case, $a=9$, $b=-39$ and $c=16$
Substitute the known values into the formula and evaluate:
$x=\dfrac{-(-39)\pm\sqrt{(-39)^{2}-4(9)(16)}}{2(9)}=...$
$...=\dfrac{39\pm\sqrt{1521-576}}{18}=\dfrac{39\pm\sqrt{945}}{18}=\dfrac{39\pm3\sqrt{105}}{18}=...$
$...=\dfrac{13}{6}\pm\dfrac{\sqrt{105}}{6}$
Verifying the solutions it can be seen that $x=\dfrac{13}{6}\pm\dfrac{\sqrt{105}}{6}$ is not a solution of the original equation.
The solutions is $x=\dfrac{13}{6}+\dfrac{\sqrt{105}}{6}$
