- A student was asked to prove by contradiction that
"there are no positive integers \(x\) and \(y\) such that \(3 x ^ { 2 } + 2 x y - y ^ { 2 } = 25\) "
The start of the student's proof is shown in the box below.
Assume that integers \(x\) and \(y\) exist such that \(3 x ^ { 2 } + 2 x y - y ^ { 2 } = 25\)
$$\Rightarrow ( 3 x - y ) ( x + y ) = 25$$
$$\begin{aligned}
& \text { If } \quad ( 3 x - y ) = 1 \quad \text { and } ( x + y ) = 25
& \left. \begin{array} { l }
3 x - y = 1
x + y = 25
\end{array} \right\} \Rightarrow 4 x = 26 \Rightarrow x = 6.5 , y = 18.5 \quad \text { Not integers }
\end{aligned}$$
Show the calculations and statements that are needed to complete the proof.