if (x-2)and(2x-1)are factors of ax^2+5x+b show that a-b=0

Since, (x-2)and(2x-1)are factors of f(x) =ax^2+5x+b

hence, f(2) = f(1/2) = 0

f(2) = a(2)^2 + 5(2) + b = 0

4a + 10 + b = 0 (1)

 

Also, f(1/2) = a(1/2)^2 + 5*1/2 + b  = 0

a + 10 + 4b = 0 (2) 

 

Subtracting 1 and 2

3a -3b = 0

a-b = 0

Hence proved.