2026 · Q15
If the roots of the equation x² + mx + n = 0 are increased by the same quantity k, then they become the roots of the equation x² + nx + m = 0. What is the value of (m+n)?
Elementary MathematicsAlgebra
- A-4
- B-2
- C2
- D4
Explanation
If α, β are the roots of x² + mx + n = 0, then α + β = −m and αβ = n. The shifted roots α + k, β + k satisfy (α + k) + (β + k) = −n and (α + k)(β + k) = m, i.e. −m + 2k = −n and n − mk + k² = m. Eliminating k = (m − n)/2 yields (n − m)(n + m + 4) = 0. Since the roots are genuinely increased (k ≠ 0), n ≠ m, forcing m + n = −4, option A.