3n^2+11n-6=0

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Solution for 3n^2+11n-6=0 equation:


Simplifying
3n2 + 11n + -6 = 0

Reorder the terms:
-6 + 11n + 3n2 = 0

Solving
-6 + 11n + 3n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-2 + 3.666666667n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 3.666666667n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 3.666666667n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 3.666666667n + n2 = 0 + 2
3.666666667n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
3.666666667n + n2 = 2

The n term is 3.666666667n.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667n + 3.361111114 + n2 = 2 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667n + n2 = 2 + 3.361111114

Combine like terms: 2 + 3.361111114 = 5.361111114
3.361111114 + 3.666666667n + n2 = 5.361111114

Factor a perfect square on the left side:
(n + 1.833333334)(n + 1.833333334) = 5.361111114

Calculate the square root of the right side: 2.315407332

Break this problem into two subproblems by setting 
(n + 1.833333334) equal to 2.315407332 and -2.315407332.

Subproblem 1

n + 1.833333334 = 2.315407332 Simplifying n + 1.833333334 = 2.315407332 Reorder the terms: 1.833333334 + n = 2.315407332 Solving 1.833333334 + n = 2.315407332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + n = 2.315407332 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = 2.315407332 + -1.833333334 n = 2.315407332 + -1.833333334 Combine like terms: 2.315407332 + -1.833333334 = 0.482073998 n = 0.482073998 Simplifying n = 0.482073998

Subproblem 2

n + 1.833333334 = -2.315407332 Simplifying n + 1.833333334 = -2.315407332 Reorder the terms: 1.833333334 + n = -2.315407332 Solving 1.833333334 + n = -2.315407332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + n = -2.315407332 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = -2.315407332 + -1.833333334 n = -2.315407332 + -1.833333334 Combine like terms: -2.315407332 + -1.833333334 = -4.148740666 n = -4.148740666 Simplifying n = -4.148740666

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.482073998, -4.148740666}

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