A positive discriminant is not the whole answer

8 written marks

Consider \((k-2)x^2+2(k-1)x+k=0\), where \(k\) is real.

1. Show that the discriminant, when the equation is quadratic, is 4.

   [2]

2. Find all \(k\) for which the original equation has two distinct real solutions. Describe what happens at the exceptional value.

   [3]

3. Show that \(x=-1\) is always a solution. For \(k\ne2\), give the other solution in terms of \(k\), and show that it never equals \(-1\).

   [3]

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