HOTS Problems : Quadratic Equation

Illustration 3. Let f (x) = Ax2 + 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) = An2 + 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 ax2 + bx + c = 0 then find the value of

(ii) If the sum of the roots of the equation ax2 + bx + c = 0 is equal to sum of the squares of their reciprocals, then show that ab2, a2c, bc2 are in A.P.

(iii) If the roots a and b of the quadratic equation ax2 + 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.

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