solving equation 6x^4+11x^3-9x^2-11x+6=0 what are the roots
Answers
Given : 6x⁴ + 11x³ - 9x² - 11x + 6 = 0
To find : Roots
Solution:
6x⁴ + 11x³ - 9x² - 11x + 6 = 0
Dividing by x²
=> 6x² + 11x - 9 - 11/x + 6/x² = 0
=> 6(x² + 1/x²) + 11(x - 1/x) - 9 = 0
=> 6(x - 1/x)² + 12 + 11(x - 1/x) - 9 = 0
=> 6(x - 1/x)² + 11(x - 1/x) + 3 = 0
Let say x - 1/x = y
=> 6y² + 11y + 3 = 0
=> 6y² + 9y + 2y + 3 = 0
=> 3y(2y + 3) + 1(2y + 3) = 0
=> (3y + 1)(2y + 3) = 0
=> y = - 1/3 , y = -3/2
x - 1/x = -1/3
=> 3x² - 3 = - x
=> 3x² + x - 3 = 0
=> x = (- 1 ± √37 )/6
x - 1/x = -3/2
=> 2x² - 2 = - 3x
=> 2x² + 3x - 2 = 0
=> 2x² + 4x - x - 2 = 0
=> 2x(x + 2) - 1(x + 2) = 0
=> (2x - 1)(x + 2) = 0
=> x = 1/2 , - 2
x = (- 1 ± √37 )/6 , 1/2 , - 2
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