Correct Answer - Option 1 :
\(\frac{{8x - {x^2}}}{{11}}\)
Calculation:
\(\sqrt {6{x^2} + 19x + 8} + \;\sqrt {7{x^2} + 11x + 8} = 11\;\;\;\;\;\; - \; - \; - \; - \left( 1 \right)\)
Let the value of \(\sqrt {6{x^2} + 19x + 8} - \;\sqrt {7{x^2} + 11x + 8} = a\;\;\;\;\; - \; - \; - \; - \left( 2 \right)\;\)
From (1) and (2)
\(\Rightarrow \left( {\sqrt {6{x^2} + 19x + 8} + \;\sqrt {7{x^2} + 11x + 8} } \right)\left( {\sqrt {6{x^2} + 19x + 8} - \;\sqrt {7{x^2} + 11x + 8} } \right) = 11a\)
\(\Rightarrow \left( {6{x^2} + 19x + 8} \right) - \left( {7{x^2} + 11x + 8} \right) = 11a\)
\(\Rightarrow - {x^2} + 8x = 11a\)
\(\Rightarrow 11a = \;8x - {x^2}\)
\(\Rightarrow a = \;\frac{{8x - {x^2}}}{{11}}\)
∴ Required answer is
\({\rm{\;}}\frac{{8x - {x^2}}}{{11}}\)
