If p and q are primes and has $x^{2}$- px+q=0 distinct positive integral roots, find p and q .

0 bình luận về “If p and q are primes and has $x^{2}$- px+q=0 distinct positive integral roots, find p and q .”

1. Đáp án:

   p+q = 2 +3 = 5 Ans.

   Giải thích các bước giải:

   Given : p and q are prime in x^2 -px +q =0
   and it has distinct positive roots
   As we know that
   for distinct roots
   D > 0
   or, (-p)^2 -4q >0
   or, p^2 – 4q >0
   or, p^2 > 4q
   or, p > √(4q)
   or, p > 2√q
   But p and q are primes and here we got p is greater than the twice of the sqrt of q.
   Let a and b be the roots of the eqn.,
   Also we can right the above equation as
   (x-a)(x-b) where a and b are distinct positive roots
   so, x^2 -ax -bx +ab =0
   or, x^2 -(a+b)x +(ab) = 0
   comparing this equation to the given eqn.
   we get (a+b) = p
   (ab) = q
   but since q is a prime no. so it can be written as (1×itself ) according to the definition of prime
   so, either a or b is equal to 1.
   hence ab = q
   or, 1×b = q
   so, b = q
   Now , as (a +b) = p
   so, 1+b = p
   or, 1+q = p
   or, p – q = 1
   And we have only one pair of such primer nos.
   whose difference is 1 and those are 2 & 3
   so, q= 2 and p = 3.
   {Also we can see that , p > 2√q
   or. 3 > 2×√2 = 2×1.414 = 2.828}
   Hence roots are distinct too.
   Now solving , p+q = 2 +3 = 5 Ans.

   Bình luận