quadratic polynomial P(x)=ax^2+bx+c has positve coefficients a,b,c in ap.P(x)=0 has integer roots p,q.Find p+q+pq

Since a, b & c are in Arithmetic Progression, b-a = c-b = Constant = K

Also p & q are the two integer roots of of the equation ax^2+bx+c = 0 — (1)

then p + q = -b/a and pq = c/a,

Now the equation can be written as x^2-(p+ q) +p q = 0 — (2)

Comparing Eq. (1) and (2) a = 1, b = -(p+ q) and c = p q

K = c - b = p q - (-p- q) = pq + p+q — (3)

K = b - a = -p - q -1 = -(p+q+1) — (4)

Therefore a = 1, b = K + 1, c = 2K +1

Now substitute the possible values of K, given in the Answers 3,5,7, 14

if K=3, x^2+4x+7, -(p+q) = 4 & pq = 7, is prime number and no integer roots.

if K=5, x^2+6x+11, -(p+q) = 6 & pq =11, is prime number and no integer roots.

If K=7, x^2+8x+15, -(p+q) = 7 & pq =15, then the roots are p= -3 & q= -5.

if K=14, x^2+15x+29, -(p+q) = 15 & pq =29, is prime number and no integer roots.

Therefore K = -p - q -1 = 3 + 5 -1 = 7,

Regards

Arun (AskIITians Forum Expert)