Question

if both (x-2)and(x-1/2)are the factors of pxsq+5x+r,prove that p=r

Answer 1

Given
$p {x}^{2} + 5x + r = 0$
(x-2) is a factor. Therefore,
$4p + r = - 10$
(x-1/2) is a factor. Therefore,
$\frac{p}{4} + \frac{5}{2} + r = 0 \\ p + 4r = - 10$
Solving
$4p + r = - 10 \\ p + 4r = - 10$
You will get p= r= -2

Answer 2

for x=2 equation is 4p+10+r=0
for x=1/2 equaion is p/4+5/2+r=0

solve simultaneously 
p=-2
r= -2

so p=r