- (i) Show that \(k ^ { 2 } - 4 k + 5\) is positive for all real values of \(k\).
(ii) A student was asked to prove by contradiction that "There are no positive integers \(x\) and \(y\) such that \(( 3 x + 2 y ) ( 2 x - 5 y ) = 28\) " The start of the student's proof is shown below.
Assume that positive integers \(x\) and \(y\) exist such that
$$\left. \begin{array} { c }
( 3 x + 2 y ) ( 2 x - 5 y ) = 28
\text { If } 3 x + 2 y = 14 \text { and } 2 x - 5 y = 2
3 x + 2 y = 14
2 x - 5 y = 2
\end{array} \right\} \Rightarrow x = \frac { 74 } { 19 } , y = \frac { 22 } { 19 } \text { Not integers }$$
Show the calculations and statements needed to complete the proof.