Question

Find the following products : (1/2xsquare +ysquare) (xsquare-1/2ysquare​

Answer 1

$( \frac{1}{2} {x}^{2} + {y}^{2} )( {x}^{2} - \frac{1}{2} {y}^{2} ) \\ \\ \frac{1}{2}{x}^{2}({x}^{2} - \frac{1}{2} {y}^{2} ) + {y}^{2} ({x}^{2} - \frac{1}{2} {y}^{2} ) \\ \\ \frac{ {x}^{4} }{2} - \frac{ {x}^{2} {y}^{2} }{4} + {x}^{2} {y}^{2} - \frac{ {y}^{4} }{2} \\ \\ \frac{ {x}^{4} }{2} - \frac{ {y}^{4} }{2} - (\frac{ {x}^{2} {y}^{2} }{4} + {x}^{2} {y}^{2}) \\ \\ \frac{ {x}^{4} }{2} - \frac{ {y}^{4} }{2} - ( \frac{ {x}^{2} {y}^{2} + {4x}^{2} {y}^{2} }{4} ) \\ \\ \frac{ {x}^{4} }{2} - \frac{ {y}^{4} }{2} -( \frac{ {5x}^{2} {y}^{2} }{4} ) \\ \\ { \boxed{ \red{\frac{ {x}^{4} }{2} - \frac{ {y}^{4} }{2} - \frac{ {5x}^{2} {y}^{2} }{4} \: \: or \: \: \frac{1}{2} {x}^{4} - \frac{1}{2} {y}^{4} - \frac{1}{4} {x}^{2} {y}^{2} }}} \\ \\ \frac{ {2x}^{4} - {2y}^{4} - {5x}^{2} {y}^{2} }{4} = { \red{ \boxed{\frac{1}{4}( {2x}^{4} - {2y}^{4} - {5x}^{2} {y}^{2} )}}}$