An algebra problem by Wildan Bagus Wicaksono

$\begin{aligned} y & = \frac {\sqrt{19-8\sqrt 3}-\sqrt{7-4\sqrt 3}}{\left(\sqrt{12}-\sqrt{18}+4\sqrt 2\right)\sqrt[4]{36}} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {\sqrt{4^2-2\cdot 4\sqrt 3 + \sqrt 3^2}-\sqrt{2^2-2\cdot 2\sqrt 3 + \sqrt 3^2}}{\left(2\sqrt{3}-3\sqrt{2}+4\sqrt 2\right)\sqrt[4]{6^2}} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {\sqrt{(4- \sqrt 3)^2}-\sqrt{(2-\sqrt 3)^2}}{\left(2\sqrt{3}+\sqrt{2}\right)\sqrt{6}} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {4- \sqrt 3-2+\sqrt 3}{6\sqrt{2}+2\sqrt 3} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {2}{6\sqrt{2}+2\sqrt 3} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {1}{3\sqrt{2}+\sqrt 3} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {3\sqrt{2}-\sqrt 3}{(3\sqrt{2}+\sqrt 3)(3\sqrt{2}-\sqrt 3)} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {3\sqrt{2}-\sqrt 3}{18-3} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \frac {3\sqrt{2}-\sqrt 3}{15} - \frac {3\sqrt 2-16\sqrt 3}{15} \\ & = \sqrt 3 \end{aligned}$

$\implies \lfloor y-1 \rfloor + 2 = \lfloor 1.732-1 \rfloor + 2 = \boxed{2}$

$\sqrt { 19-8\sqrt { 3 } } =\sqrt { 19-2\sqrt { 16\cdot 3 } } =\sqrt { { \left( 4-\sqrt { 3 } \right) }^{ 2 } } =4-\sqrt { 3 } \\ \sqrt { 7-4\sqrt { 3 } } =\sqrt { 7-2\sqrt { 4\cdot 3 } } =\sqrt { { \left( 2-\sqrt { 3 } \right) }^{ 2 } } =2-\sqrt { 3 } \\ \sqrt { 19-8\sqrt { 3 } } -\sqrt { 7-4\sqrt { 3 } } =\left( 4-\sqrt { 3 } \right) -\left( 2-\sqrt { 3 } \right) \\ \sqrt { 19-8\sqrt { 3 } } -\sqrt { 7-4\sqrt { 3 } } =4-\sqrt { 3 } -2+\sqrt { 3 } \\ \sqrt { 19-8\sqrt { 3 } } -\sqrt { 7-4\sqrt { 3 } } =2$

$y=\frac { 2 }{ \left( \sqrt { 12 } -\sqrt { 18 } +4\sqrt { 2 } \right) \sqrt [ 4 ]{ 36 } } -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 }$

$=\frac { 2 }{ \left( \sqrt { 12 } -\sqrt { 18 } +4\sqrt { 2 } \right) \sqrt [ 4 ]{ 36 } } -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 } \\ =\frac { 2 }{ \left( 2\sqrt { 3 } -3\sqrt { 2 } +4\sqrt { 2 } \right) \sqrt { 6 } } -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 } \\ =\frac { 2 }{ \left( 2\sqrt { 3 } +\sqrt { 2 } \right) \sqrt { 6 } } -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 } \\ =\frac { 2 }{ 6\sqrt { 2 } +2\sqrt { 3 } } -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 } \\ =\frac { 1 }{ 3\sqrt { 2 } +\sqrt { 3 } } \cdot \left( \frac { 3\sqrt { 2 } -\sqrt { 3 } }{ 3\sqrt { 2 } -\sqrt { 3 } } \right) -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 } \\ =\frac { 3\sqrt { 2 } -\sqrt { 3 } }{ 18-3 } -\frac { 3\sqrt { 2 } -16\sqrt { 3 } }{ 15 } \\ =\frac { 3\sqrt { 2 } -\sqrt { 3 } -3\sqrt { 2 } +16\sqrt { 3 } }{ 15 } \\ =\frac { 15\sqrt { 3 } }{ 15 } \\ =\sqrt { 3 }$

$y=\sqrt { 3 } \\ \left\lfloor y-1 \right\rfloor +2=\left\lfloor \sqrt { 3 } -1 \right\rfloor +2\\ \left\lfloor y-1 \right\rfloor +2=\left\lfloor 1....-1 \right\rfloor +2\\ \left\lfloor y-1 \right\rfloor +2=\left\lfloor 0.... \right\rfloor +2\\ \left\lfloor y-1 \right\rfloor +2=0+2\\ \left\lfloor y-1 \right\rfloor +2=2$ .