2026 · Q19
Let Q = x^2 + bx + c. If the sum of the roots is equal to product of the roots of the equation Q = 0, then which of the following statements is/are correct?
- IQ can be a perfect square.
- IIQ is positive for all real values of x. Select the answer using the code given below:
Elementary MathematicsAlgebra
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
For Q=x²+bx+c the sum of roots is −b and the product is c; equality gives c=−b. Statement I: with b=0, Q=x² is a perfect square, so Q can indeed be one — I is correct. Statement II: Q≥0 for all x would need discriminant b²+4b<0 for every admissible b, but b=0 gives Q=x² which is zero at x=0 — II is incorrect. Only I is correct, option A.