If m is chosen in the quadratic equation (m² + 1)x² - 3x + (m² + 1)² = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

Correct option is D. 85
SOR = 3/(m² + 1) ⇒ (S.O.R)max = when m = 0
α + β = 3; αβ = 1
|α³ - β³| = |α - β||(α² + β² + αβ)| = |(α - β)² - αβ|(α - β)² - αβ|
α + β = 3
αβ = 1
(α + β)² = α² + β² + 2αβ
9 = α² + β² + 2(1)
α² + β² = 7
|α³ - β³| = |(α - β)(α² + αβ + β²)| = |(α - β)(7 + 1)| = 8|α - β|
(α - β)² = (α + β)² - 4αβ = 9 - 4 = 5
|α - β| = √5
|α³ - β³| = 8√5
This value is not among the options. Let's re-examine.
Sum of roots = 3/(m²+1). This is maximum when m=0. Then the equation becomes x²-3x+1=0.
Roots are α, β such that α+β=3 and αβ=1.
|α³-β³| = |(α-β)(α²+αβ+β²)| = |(α-β)((α+β)²-αβ)| = |(α-β)(9-1)| = 8|α-β|
(α-β)² = (α+β)²-4αβ = 9-4 = 5
|α-β| = √5
Then |α³-β³| = 8√5 ≈ 17.88
Let's use the quadratic formula to find the roots when m=0:
x = (3 ± √(9 - 4))/2 = (3 ± √5)/2
α = (3 + √5)/2, β = (3 - √5)/2
α³ - β³ = [(3 + √5)/2]³ - [(3 - √5)/2]³ = (27 + 27√5 + 45 + 5√5)/8 - (27 - 27√5 + 45 - 5√5)/8 = (54√5 + 54)/8 = 27(√5 + 1)/4 ≈ 17.888
There must be an error in the question or options.