Given that \(2 ^ { x } = \frac { 1 } { \sqrt { 2 } }\) and \(2 ^ { y } = 4 \sqrt { } 2\),
find the exact value of \(x\) and the exact value of \(y\),
calculate the exact value of \(2 ^ { y - x }\).
(a) Prove, by completing the square, that the roots of the equation \(x ^ { 2 } + 2 k x + c = 0\), where \(k\) and \(c\) are constants, are \(- k \pm v ^ { 2 } - c\) ).
The equation \(x ^ { 2 } + 2 k x \pm 81 = 0\) has equal roots.