2. (a) For what values of the constant $k$ does the quadratic equation $$x ^ { 2 } - 2 x - 1 = k$$ have: (i) no real solutions; (ii) one real solution; (iii) two real solutions. (b) Showing your working, express $\left( x ^ { 2 } - 2 x - 1 \right) ^ { 2 }$ as a polynomial of degree 4 in $x$. (c) Show that the quartic equation $$x ^ { 4 } - 4 x ^ { 3 } + 2 x ^ { 2 } + 4 x + 1 = h$$ has exactly two real solutions if either $h = 0$ or $h > 4$. Show that there is no value of $h$ such that the above quartic equation has just one real solution.
2. (a) For what values of the constant $k$ does the quadratic equation
$$x ^ { 2 } - 2 x - 1 = k$$
have:\\
(i) no real solutions;\\
(ii) one real solution;\\
(iii) two real solutions.\\
(b) Showing your working, express $\left( x ^ { 2 } - 2 x - 1 \right) ^ { 2 }$ as a polynomial of degree 4 in $x$.\\
(c) Show that the quartic equation
$$x ^ { 4 } - 4 x ^ { 3 } + 2 x ^ { 2 } + 4 x + 1 = h$$
has exactly two real solutions if either $h = 0$ or $h > 4$.\\
Show that there is no value of $h$ such that the above quartic equation has just one real solution.\\