Step-by-step explanation:
(λ + 1) x² − 2x + (1 − λ) = 0
The discriminant is:
D = b² − 4ac
D = (-2)² − 4 (λ + 1) (1 − λ)
D = 4 − 4 (λ − λ² + 1 − λ)
D = 4 − 4 (1 − λ²)
D = 4 − 4 + 4λ²
D = 4λ²
Since D ≥ 0, the roots are real.
Using quadratic formula, the roots are:
x = [ -b ± √(b² − 4ac) ] / 2a
x = (2 ± 2λ) / (2(λ + 1))
x = (1 ± λ) / (λ + 1)
α, β = 1 or (1 − λ) / (λ + 1)
Therefore, the squares of the roots are:
α², β² = 1 or (1 − λ)² / (λ + 1)²
And the equation with these roots is:
(x − 1) (x − (1 − λ)² / (λ + 1)²) = 0
(x − 1) ((λ + 1)² x − (1 − λ)²) = 0
(λ + 1)² x² − (1 − λ)² x − (λ + 1)² x + (1 − λ)² = 0
(λ + 1)² x² − (1 − λ + λ + 1)² x + (1 − λ)² = 0
(λ + 1)² x² − 4x + (1 − λ)² = 0
