**Illustration 3.** Let f (x) = Ax^{2} + Bx + C where A, B, C are real numbers. Prove that if f (x) is an integer whenever x is an integer, then the numbers 2A, A + B and C are all integers. Conversely, prove that if the numbers 2A, A + B and C are all integer then f (x) is an integer whenever x is an integer.

**Solution:** Let us consider the integral values of x as 0 , 1 , − 1 then f (0) , f (1) and f (- 1) are all integers => f (0) = C, f (1) = A + B + C and f (- 1) = A – B + C all integers

Therefore C is an integer and hence A + B is an integer also A − B is an integer

Now, 2A = (A + B) + (A – B) = sum of two integers

=> 2A is also an integer

=> 2A , A + B and C are all integers

conversely let n ∈ I then f (n) = An^{2} + Bn + C

= 2A[n(n − 1)/2] + (A + B) n + C

now n(n − 1)/2 = even/2 = integer and 2A, A + B and C are also integer

=> f (x) is an integer

**Exercise 1: **

**(i)** If sinθ, cosθ are the roots of the equation ax^{2} + bx + c = 0 then find the value of

**(ii)** If the sum of the roots of the equation ax^{2} + bx + c = 0 is equal to sum of the squares of their reciprocals, then show that ab^{2}, a^{2}c, bc^{2} are in A.P.

**(iii)** If the roots a and b of the quadratic equation ax^{2} + bx + c = 0 are real and of opposite sign. Then show that roots of the equation α(x − β)^{2} + β(x − α)^{2} = 0 are also real and of opposite sign.