The Square Root Of 11 Derivation

x=\sqrt { 11 } \\ \\ { x }^{ 2 }=11\\ \\ { x }^{ 2 }+3x=11+3x\\ \\ x\left( 3+x \right) =3\left( 3+x \right) +2\\ \\ \frac { x\left( 3+x \right) }{ \left( 3+x \right) } =\frac { 3\left( 3+x \right) }{ \left( 3+x \right) } +\frac { 2 }{ \left( 3+x \right) } \\ \\ x=3+\frac { 2 }{ 3+x } \\ \\ \sqrt { 11 } =3+\frac { 2 }{ 3+x } \\ \\ =3+\frac { 2 }{ 3+\left( 3+\frac { 2 }{ 3+x } \right) }

\\ \\ =3+\frac { 2 }{ 6+\frac { 2 }{ 3+x } } \\ \\ =3+\frac { 2 }{ 6+\frac { 2 }{ 3+\left( 3+\frac { 2 }{ 3+x } \right) } } \\ \\ =3+\frac { 2 }{ 6+\frac { 2 }{ 6+\frac { 2 }{ 3+x } } } \\ \\ \therefore \sqrt { 11 } =3+\frac { 2 }{ 6+\frac { 2 }{ 6+\frac { 2 }{ 6+\frac { 2 }{ 6+... } } } }

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