Question

If x is equal to under root 5 + under root 3 upon under root 5 minus under root 3 and Y is equal to under root 5 minus under root 3 upon under root 5 + under root 3 find the value of x square + Y square

Answer 1

Answer:

$x \\ = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \\ = \frac{( \sqrt{5} + \sqrt{3}) {}^{2} }{( \sqrt{5} - \sqrt{3})( \sqrt{5} + \sqrt{3}) } \\ = \frac{ (\sqrt{5}) {}^{2} + (\sqrt{3}) {}^{2} + 2 \sqrt{5} \times \sqrt{3} }{( \sqrt{5})^{2} - ( \sqrt{3}) {}^{2} } \\ = \frac{5 + 3 + 2 \sqrt{15} }{5 - 3} \\ = \frac{8 + 2 \sqrt{15} }{2} \\ = 4 + \sqrt{15}$

$y \\ = \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ = \frac{( \sqrt{5} - \sqrt{3} ) {}^{2} }{( \sqrt{5} + \sqrt{3} )( \sqrt{5} - \sqrt{3}) } \\ = \frac{( \sqrt{5} ) {}^{2} + (\sqrt{3}) {}^{2} - 2 \sqrt{5} \times \sqrt{3} }{( \sqrt{5}) {}^{2} - ( \sqrt{3} ) {}^{2} } \\ = \frac{5 + 3 - 2 \sqrt{15} }{5 - 3} \\ = \frac{8 - 2 \sqrt{15} }{2} \\ = 4 - \sqrt{15}$

now,

${x}^{2} + {y}^{2} \\ = (4 + 2\sqrt{15} ) {}^{2} + (4 - 2 \sqrt{15} ) {}^{2} \\ = 2( {4}^{2} + (2 \sqrt{15} ) {}^{2} ) \\ = 2(16 + 60) \\ = 2 \times 76 \\ = 152$