let c is the common root then c will stisfy both of these equations .
x2-3x-k = 0                 .............1
4x2-11x+2k = 0              ............2
 
at x = c ,eq 1 becomes
c2-3c-k = 0
2c2 - 6c - 2k = 0        ..............3         (after multiplying  by 2)
at x = c, eq 2 becomes
4c2-11c+2k = 0             ................4
 
adding 3 & 4
6c2 -17c = 0
 c = 0 , 17/6
now , put c in any eq 3 or 4
then , k = 0,-17/36
 
(c,k) = { (0,0) ,(17/6,-17/36) }
this is the required solution ...
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Take both as equations

x2-3x-k = 0                 .............1
4x2-11x+2k = 0              ............2

 Solve it by eliminating method:
2x2-6x-2k = 0  [multiply by 2}........(3)
now from eq. (2) and (3)
4x2-11x+2k = 0 
2x2-6x-2k = 0
Solving these equations by eliminating method u will get:
6x2-17x = 0 .....................(4)
x(6x-17) = 0
6x = 17
x = 0, 17/6
Put the value of x in eq...........(4)
k = 0,-17/36
then, 
(c,k) = { (0,0) ,(17/6,-17/36) }
 
 
						

							
x2 -4x+k = 0 and x2+kx-4 =0 have a common root, so let’s take it y
putting value of y
y2-4y+k=0 ----{1}
y2-ky+4=0 ------{2}
substracting {1} from {2}
y2-4y+k-[ y2-ky+4]
y2-4y+k-y2+ky-4=0
-y(k+4)+4+k=0
4+k= y(k+4)
y=1
putting y=1 in eq {1}
y2-4y+k=0
(1)2-4(1)+k=0
1-4+k=0
k=3