Question

If p(x)=ax^2+bx+c=0.and it has positive coefficients.and a,b,c are in arithmetic progression.If p,q are roots of the equation, then find (p+q+pq).

Answer 1

If the given quadratic has integer roots α,β then

ax2+bx+c{ba=−(α+β)ca=αβ=a(x2+bax+ca)=a(x−α)(x−β)=a[x2−(α+β)x+αβ]=0

Given that a,b,ca,b,c are in AP, then 1,ba,caare also in AP.

The problem reduces to finding α+β+αβwhere 1,−(α+β),αβare positive integers in AP.

For the three numbers to be positive, α,β<0(α,β∈Z)

For them to be in AP,

−2(α+β)β=1+αβ=−2+3α+2

Since β is an integer, 3α+2must be an integer.

As α,β<0and α,β∈Zpossible values for α,β are (−3,−5),(−5,−3).

Hence

α+β+αβ=7