2026 · Q4
Suppose p and q are positive integers where q is not a perfect square. If 1/(p+√q) is a root of a quadratic equation having integer coefficients, then which of the following statements is/are correct?
- IThe sum of the roots is 2p times the product of the roots.
- IIThe sum of the reciprocals of the roots is equal to 2p. Select the answer using the code given below:
Elementary MathematicsAlgebra
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
If 1/(p + √q) is a root of a quadratic with integer coefficients, then since q is not a perfect square (√q is irrational), the other root must be its conjugate 1/(p − √q). The sum of the roots is 2p/(p² − q) and the product is 1/(p² − q), so the sum equals 2p times the product — statement I is correct. The sum of the reciprocals is (p + √q) + (p − √q) = 2p — statement II is also correct. Hence both statements hold, option C.