Mathematics

Problem

Consider the quadratic polynomial

h(x)=x^{2}+2(k-3)x+k+5

, where

k

is a real parameter. a. (i) Show that the number of real zeros of

h(x)

depends on the value of

k

. b. (ii) Find the set of all real values of

k

for which: (a)

h(x)

has two distinct real zeros, (b)

h(x)

has exactly one real zero, (c)

h(x)

has no real zeros. c. (iii) For each of the cases (a), (b) and (c) in part (ii), draw a possible rough sketch of the graph of

y=h(x)

to illustrate your answer.