Question

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

Answer 1

$put \: x = 2 \\ 4p + r + 10= 0 \\ 4p + r = - 10 \: \: \: eq1 \\ put \: x = \frac{1}{2} \\ \frac{p}{4} + \frac{5}{2} + r = 0 \\ \frac{p + 10 + 4r}{4} = 0 \\ p + 4r = - 10 \: \: \: eq2 \\ multiply \: eq \: 2 \: by \: 4 \: and \\ \: subtract \: both \: eq1 \: and \: 2 \\ 4p + 16r = - 40 \\ 4p + r = - 10 \\ 15r = - 30 \\ r = - 2 \: \: \: \: eq3 \\ 4p - 2 = - 10 \\ 4p = - 10 + 2 \\ 4p = - 8 \\ p = - 2 \: \: \: \: eq4 \\ from \: both \: eq \: 3 \: and \: 4 \: we \: say \: that\\ p = r = - 2 \\ p = r \\ hence \: proved$