Question

1x -2 and x -1 /2 are factors of px square+5x +r , then show that p=r

Answer 1

px^2 + 5x + r

put x= 2 and equate to 0

4p +10 + r = 0

put x = 1/2 and equate to 0

p/4 + 5/2 + r =0

p + 10 + 4r = 0

Also
4p +10 + r = 0

16 p + 40 + 4r = 0

and p + 10 +4r = 0

subtract

15p + 30 = 0

p = -2

put in p +10 +4r = 0

-2 +10 +4r = 0

4 r = -8

r = -2

So p= r = -2

Answer 2

let p(x)=px²+5x+r,

since x-2 is a factor of p(x), then
p(2)=0,

p(2)²+5(2)+r=0,

4p+10+r=0,

4p+r=-10,..........eq(1),


also x-1/2 is a factor of p(x), then

p(1/2) = 0,

p × (1/2)² + 5×1/2 + r=0,

p/4 + 5/2 + r = 0,

(p+10+4r)/4 = 0,

then

p+4r = -10,.......eq(2),

now from eq(1) and eq(2), We get

4p+r = p+4r,

4p-p = 4r-r,

3p = 3r,

then

p=r