Question

If x+y = 3, x×y = 2 find x2 - y2

Answer 1

$xy = 2 \\ \\ x + y = 3 \\ \\ squaring \: on \: both \: sides \\ \\ {(x + y)}^{2} = {(3)}^{2} \\ {x}^{2} + {y}^{2} + 2xy = 9 \\ {x}^{2} + {y}^{2} + 2(2) = 9 \\ {x}^{2} + {y}^{2} + 4 = 9 \\ {x}^{2} + {y}^{2} = 9 - 4 \\ {x}^{2} + {y}^{2} = 5 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: (i) \\ \\ {( x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} = {x}^{2} + {y}^{2} - 2xy \\ {(x - y)}^{2} = 5 - 2(2) \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: (from \: i) \\ {(x - y)}^{2} = 5 - 4 \\ {(x - y)}^{2} = 1 \\ {x - y} = 1 \: \: \: \: \: ... \: (taking \: square \: roots \: o n \: both \: sides) \\ \\ now \\ \: \: \: \: {x}^{2} - {y}^{2} \\ = (x + y)(x - y) \\ = (3)(1) \\ = 3$

Hence, x² - y² = 3