# Square Root Of 5 Derivation

$x=\sqrt { 5 } \\ \\ { x }^{ 2 }=5\\ \\ { x }^{ 2 }+2x=5+2x\\ \\ x\left( 2+x \right) =2\left( 2+x \right) +1\\ \\ \frac { x\left( 2+x \right) }{ \left( 2+x \right) } =\frac { 2\left( 2+x \right) }{ \left( 2+x \right) } +\frac { 1 }{ \left( 2+x \right) } \\ \\ x=2+\frac { 1 }{ 2+x } \\ \\ \sqrt { 5 } =2+\frac { 1 }{ 2+x } \\ \\ =2+\frac { 1 }{ 2+\left( 2+\frac { 1 }{ 2+x } \right) }$

$\\ \\ =2+\frac { 1 }{ 4+\frac { 1 }{ 2+x } } \\ \\ =2+\frac { 1 }{ 4+\frac { 1 }{ 2+\left( 2+\frac { 1 }{ 2+x } \right) } } \\ \\ =2+\frac { 1 }{ 4+\frac { 1 }{ 4+\frac { 1 }{ 2+x } } } \\ \\ \therefore \sqrt { 5 } =2+\frac { 1 }{ 4+\frac { 1 }{ 4+\frac { 1 }{ 4+... } } }$