Show that the quadratic equation \({x^2} - (5 - k)x - (k + 2) = 0\) has two distinct real roots for all real values of k .

\(\Delta  = {(5 - k)^2} + 4(k + 2)\)     M1A1

\( = {k^2} - 6k + 33\)     (A1)

\( = {(k - 3)^2} + 24\) which is positive for all k     R1

Note: Accept analytical, graphical or other correct methods. In all cases only award R1 if a reason is given in words or graphically. Award M1A1A0R1 if mistakes are made in the simplification but the argument given is correct.

[4 marks]

Overall the question was pretty well answered but some candidates seemed to have mixed up the terms determinant with discriminant. In some cases a lack of quality mathematical reasoning and understanding of the discriminant was evident. Many worked with the quadratic formula rather than just the discriminant, conveying a lack of understanding of the strategy required. Errors in algebraic simplification (expanding terms involving negative signs) prevented many candidates from scoring well in this question. Many candidates were not able to give a clear reason why the quadratic has always two distinct real solutions; in some cases a vague explanation was given, often referring to a graph which was not sketched.