We have
#f(x)=kx^2-4kx+16#
The roots of the equation are given by the discriminant #Delta#:
#Delta = b^2-4ac#
#{(x_1 =(-b+sqrtDelta)/(2a)),(x_2=(-b-sqrtDelta)/(2a)) :}#
As such, the conditions of existence of #x_1# and #x_2# are determined by #Delta#.
#diamond Delta>0 " there are two distinct real solutions"#
#diamond Delta=0 " there is one solution"#
#diamond Delta<0 " there are no solutions in real numbers"#
#Delta = 16k^2-64k#
#cc a)" f(x) has no real roots"#
#:. Delta < 0 => 16k^2-64k<0#
For a quadratic function #phi# with leading coefficient positive and roots #r_1#, #r_2#, #r_1>r_2#, the equation
#phi(x)<0#
has solutions in the interval
#x in (r_2,r_1)#
In our case,
#phi(k) = 16k^2-64k=16k(k-4)#
#{(r_1 = 4), (r_2 = 0) :}#
#phi(k)< 0 => color(red)(k in (0,4)#
#cc b)" f(x) has two equal roots"#
#:. Delta = 0 => 16k^2-64k=0 => color(red)(k in {0,4}#
#cc c)" f(x) has two distinct roots"#
#:. Delta > 0 => 16k^2-64k>0#
Similarly, the equation
#phi(x)>0#
where #phi# is the same quadratic function has solutions
#x in (-oo, r_2) uu (r_1, +oo)#
#phi(k) > 0 => 16k^2-64k > 0 => color(red)(k in (-oo, 0) uu (4, + oo)#
