Find the integer pair (a,b) that satisfies this constraint: x2−ax−b=0 has integer solutions and x2+ax+b=0 has integer solutions. The pair (5,6) satisfies this because x2−5x−6=0 has integer solutions i.e. x={−1,6} and x2+5x+6=0 has integer solutions i.e. x={−3,−2}.If there is no general form can you help me to at least find another pair of integers that satisfies this constraint other than (5,6)? Thanks

Question

annieljcddj0 · Accepted Answer

Let the discriminants of the two equations be a2+4b=k2 and a2−4b=l2. Note that- a,k,l have the same parity, so the discriminants being perfect squares is enough to ensure integer solutions- the difference of two squares of the same parity is always a multiple of 4, so we need not worry about b being non-integral – it suffices to get a,k,lNow l2,a2,k2 are perfect squares in arithmetic progression, so l2+k2=2a2 or (la)2+(ka)2=2. All rational solutions for l/a and k/a may be parametrised as follows:la=m2−2m−1m2+1ka=m2+2m−1m2+1where m∈Q∪{∞}.Given a rational solution for la and ka, write the fractions with a common denominator, from which l,a,k are immediately determined up to sign (their signs must all agree). From there b may be easily worked out.

legaldaj1dn · Accepted Answer

The product of the roots of one equation must be the negative of the product of the other pair of roots. Also, the sum of the roots of one equation must be the negative of the sum of the other pair of roots.A formula for this is to let the pairs of roots beL2−3LM+2M2,LM and M(L−2M),L(M−L) for integers L and M.

Find the integer pair (a,b) that satisfies this

Answered question

Answer & Explanation

New Questions in Algebra I