For what values of a ∈ Z , the quadratic expression x + a x + 1991 + 1 can be factorised as x + b x + c , where b , c ∈ Z ?

Step-by-step solution

Since, he quadratic expression x + a x + 1991 + 1 can be factorised as x + b x + c , therefore x + a x + 1991 + 1 = x + b x + c This implies that x + a x + 1991 + 1 has roots - b   &   - c i.e., integer roots. So, discriminant of x 2 + 1991 + a x + 1991 a + 1 = 0 must be perfect square. 1991 + a 2 - 4 1991 a + 1 = m 2 ⇒ 1991 - a 2 = m 2 + 4 ⇒ 1991 - a 2 - m 2 = 4 ⇒ 1991 - a - m 1991 - a + m = 4 Since, both 1991 - a - m   and 1991 - a + m are integers, therefore possible choices