[中四數]Quadratic Equations in One Unknown[20分]

(a) discriminant = b^2 -4ac
= (-(2k+1))^2 -4 x (k+2) x (k-3)
= (-2k-1)^2 -4 x (k+2) x (k-3)
= 4k^2+2k+1-4 (k^2 -k -6)
= 4k^2+2k+1-4 k^2 -4k -24
= -2k -23

(b)
(i) two unequal real roots===>discriminant>0
-2k -23>0
-2k>23
k<23/2

(ii)two equal real roots===>discriminant=0
-2k -23=0
k=23/2

(iii)no real root===>discriminant<0
-2k -23<0
-2k<23
k>23/2