If $x = \sqrt{10} + \sqrt{11}, y = \sqrt{10} - \sqrt{11}, $ then value of $ 7x^2 - 50 xy + 7y^2$ = ____________.

344

Formula used,

( a - b )² = a² + b² - 2ab

( a + b )² = a² + b² + 2ab

a² - b² = (a + b) (a – b)

If $x = \sqrt{10} + \sqrt{11}, y = \sqrt{10} - \sqrt{11}, $

then value of $ 7x^2 - 50 xy + 7y^2$

The value of $x = \sqrt{10} + \sqrt{11}$

x² = (\(\sqrt {10}\))² + (\(\sqrt {11}\))² + 2(\(\sqrt {10}\))(\(\sqrt {11}\))

x² =10 + 11 + 2(\(\sqrt {110}\))

x² = 21 + 2(\(\sqrt {110}\))

and now the value of y² will be,

y² = 21 - 2(\(\sqrt {110}\))

and xy = ($\sqrt{11} + \sqrt{10}$)($\sqrt{11} - \sqrt{10}$) = 1

Put these values in the required equation,

$ 7x^2 - 50 xy + 7y^2$ =  7(21 + 2(\(\sqrt {110}\))) - 50 (1) + 7(21 - 2(\(\sqrt {110}\)))

$ 7x^2 - 50 xy + 7y^2$ = 147 + 14(\(\sqrt {110}\)) + 147 - 14(\(\sqrt {110}\)) + 50

$ 7x^2 - 50 xy + 7y^2$ = 344