The mean proportion between $$\left( {3 + \sqrt 2 } \right)$$   and $$\left( {12 - \sqrt {32} } \right)$$   is = ?

$$\eqalign{ & \left( {3 + \sqrt 2 } \right):x:\left( {12 - \sqrt {32} } \right) \cr & a:b:c \cr & {\text{mean proportion}} \cr & \Rightarrow {b^2} = a \times c \cr & \Rightarrow {x^2} = \left( {3 + \sqrt 2 } \right) \times \left( {12 - \sqrt {32} } \right) \cr & \Rightarrow {x^2} = \left( {3 + \sqrt 2 } \right) \times \left( {12 - 4\sqrt 2 } \right) \cr & \Rightarrow {x^2} = \left( {3 + \sqrt 2 } \right) \times 4\left( {3 - \sqrt 2 } \right) \cr & \Rightarrow {x^2} = 4 \times \left\{ {{{\left( 3 \right)}^2} - {{\left( {\sqrt 2 } \right)}^2}} \right\} \cr & \Rightarrow {x^2} = 28 \cr & \Rightarrow x = 2\sqrt 7 \cr} $$