Explanation

If the two quadratics share a common factor, some root r satisfies both. Subtracting the equations eliminates x²: (x²−5x+α)−(x²−7x+2α) = 2x−α = 0, so the common root is r = α/2. Substituting r into x²−5x+α gives (α/2)²−5(α/2)+α = α²/4−3α/2 = α(α−6)/4 = 0. Since α≠0, α = 6. Hence option (A) 6.