The equation \({x^2} + (k + 2)x + 2k = 0\) has two distinct real roots.

Find the possible values of \(k\).

evidence of discriminant     (M1)

eg     \({b^2} - 4ac,{\text{ }}\Delta  = 0\)

correct substitution into discriminant     (A1)

eg     \({(k + 2)^2} - 4(2k),{\text{ }}{k^2} + 4k + 4 - 8k\)

correct discriminant     A1

eg     \({k^2} - 4k + 4,{\text{ }}{(k - 2)^2}\)

recognizing discriminant is positive     R1

eg     \(\Delta  > 0,{\text{ }}{(k + 2)^2} - 4(2k) > 0\)

attempt to solve their quadratic in \(k\)     (M1)

eg     factorizing, \(k = \frac{{4 \pm \sqrt {16 - 16} }}{2}\)

correct working     A1

eg     \({(k - 2)^2} > 0,{\text{ }}k = 2\), sketch of positive parabola on the x-axis

correct values     A2     N4

eg     \(k \in \mathbb{R}{\text{ and }}k \ne 2,{\text{ }}\mathbb{R}\backslash 2,{\text{ }}\left] { - \infty ,{\text{ }}2} \right[ \cup \left] {2,{\text{ }}\infty } \right[\)

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