For what value(s) of `lamda` the quadratic equation `x^(2)+lamdax+(lamda+1.25)=0` has :
(i) coincident roots (ii) real and distinct roots (iii) no real roots?

Explanation

We have
`x^(2)+lamdax+(lamda+1.25)=0`
Campare it with `ax^(2)+bx+c=0`, we get `:.a=1, b=lamda,c=lamda+1.25`
`:.D=b^(2)-4ac=lamda^(2)-4(1)(lamda+1.25)`
`impliesD=lamda^(2)-4lamda-5=(lamda-5)(lamda+1)`
(i) For coincident roots,
D=0
`implies(lamda-5)(lamda+1)=0`
`:.lamda=5orlamda=-1`
(ii) Now, two cases may be there
Case I: `lamda-5gt0andlamda+1gt0" "[:'(+)(+)=(+)]`
`implieslamdagt5andlamdagt-1`
`implieslamdagt5" ".....(1)`
Case II : `lamda-5lt0andlamda+1lt0" "[:'(-)(-)=(+)]`
`lamdalt5andlamdalt-1`
`implieslamdalt-1" ".....(2)`
from (1) and (2), we have
`lamdagt5orlamdalt-1`
(iii) For no real (imaginary) roots,
`Dlt0`
`implies(lamda-5)(lamda+1)lt0`
Now two cases may arise.
Case I : `lamda-5gt0andlamda+1gt0" "[:'(+)(-)=-]`
`implieslamdalt5andlamdalt-1`
which is not possible (as there is no value which is greater than 5 as well as less than-1 also)
Case II : `lamda-5lt0andlamda+1gt0" "[:'(-)(+)=-]`
`implieslamdaltandlamdagt-1`
`implies-1ltlamdalt5i.e.,lamda` lies between -1 and 5.