7 The curve \(C\) has equation \(y = x ^ { 2 } + 7\). The line \(L\) has equation \(y = k ( 3 x + 1 )\), where \(k\) is a constant.
- Show that the \(x\)-coordinates of any points of intersection of the line \(L\) with the curve \(C\) satisfy the equation
$$x ^ { 2 } - 3 k x + 7 - k = 0$$
- The curve \(C\) and the line \(L\) intersect in two distinct points. Show that
$$9 k ^ { 2 } + 4 k - 28 > 0$$
- Solve the inequality \(9 k ^ { 2 } + 4 k - 28 > 0\).