if two roots of 2x^3-11x^2+ 12x+9=0 are equal ,then the roots are
Answers
Given : two roots of 2x^3-11x^2+ 12x+9=0 are equal
To Find : roots
Solution:
2x³ - 11x² + 12x + 9 = 0
Let say roots are α , α , β
Sum of roots 2 α + β = -(-11/2) = 11/2
=> β = (11/2 - 2 α)
Products of roots = α²β = -9/2
α² + 2αβ = 12/2
=> α² + 2αβ = 6
=> α² + 2α (11/2 - 2 α) = 6
=> α² + 11α - 4α² = 6
=> 3α² - 11α + 6 = 0
=> 3α² - 9α - 2α + 6 = 0
=> 3α(α - 3) - 2(α - 3) = 0
=> (α - 3)(3α - 2) = 0
=> α = 3 , 2/3
α = 3 , β = -1/2
α = 2/3 , β = 29/6 ( not possible as α²β is - ve )
roots are 3 , 3 , - 1/2
2x³ - 11x² + 12x + 9 = 0 = (x - 3)²(2x + 1)
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