3n^2-11n-10=0

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Solution for 3n^2-11n-10=0 equation: 


Simplifying
3n2 + -11n + -10 = 0

Reorder the terms:
-10 + -11n + 3n2 = 0

Solving
-10 + -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'.
-3.333333333 + -3.666666667n + n2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -3.666666667n + 3.333333333 + n2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -3.666666667n + n2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -3.666666667n + n2 = 0 + 3.333333333
-3.666666667n + n2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-3.666666667n + n2 = 3.333333333

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 = 3.333333333 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667n + n2 = 3.333333333 + 3.361111114

Combine like terms: 3.333333333 + 3.361111114 = 6.694444447
3.361111114 + -3.666666667n + n2 = 6.694444447

Factor a perfect square on the left side:
(n + -1.833333334)(n + -1.833333334) = 6.694444447

Calculate the square root of the right side: 2.58736245

Break this problem into two subproblems by setting 
(n + -1.833333334) equal to 2.58736245 and -2.58736245.


Subproblem 1
n + -1.833333334 = 2.58736245

Simplifying
n + -1.833333334 = 2.58736245

Reorder the terms:
-1.833333334 + n = 2.58736245

Solving
-1.833333334 + n = 2.58736245

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 = 2.58736245 + 1.833333334

Combine like terms: -1.833333334 + 1.833333334 = 0.000000000
0.000000000 + n = 2.58736245 + 1.833333334
n = 2.58736245 + 1.833333334

Combine like terms: 2.58736245 + 1.833333334 = 4.420695784
n = 4.420695784

Simplifying
n = 4.420695784


Subproblem 2
n + -1.833333334 = -2.58736245

Simplifying
n + -1.833333334 = -2.58736245

Reorder the terms:
-1.833333334 + n = -2.58736245

Solving
-1.833333334 + n = -2.58736245

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 = -2.58736245 + 1.833333334

Combine like terms: -1.833333334 + 1.833333334 = 0.000000000
0.000000000 + n = -2.58736245 + 1.833333334
n = -2.58736245 + 1.833333334

Combine like terms: -2.58736245 + 1.833333334 = -0.754029116
n = -0.754029116

Simplifying
n = -0.754029116


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.420695784, -0.754029116}

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