Question

if x-2 and (x-1/2) are both factors of px^2+5x+r show that p=r

Answer 1

If the x - 2 and x - 1/2 are the factors then the value of x will be 2 and 1/2.
Now,
On putting x = 2 in the following equ.
p* 2^2 + 5*2 + r = 0
=> 4p + 10 + r = 0
=> 4p + r = -10. equ. 1
Then,
Put x = 1/2
Therefore,
p * 1/2*1/2 + 5*1/2 + r = 0
=> p/4 + 5/2 + r = 0
=> p + 10 + 4r / 4 = 0
=> p + 4r = -10. equ. 2
On subtracting equ. 1st and 2nd, we get;
4p + r - p - 4r = -10 + 10
=> 3p - 3r = 0
=> 3(p - r) = 0
=> p - r = 0
Hence, p = r.