126=6n^2+n-1

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Solution for 126=6n^2+n-1 equation:


Simplifying
126 = 6n2 + n + -1

Reorder the terms:
126 = -1 + n + 6n2

Solving
126 = -1 + n + 6n2

Solving for variable 'n'.

Combine like terms: 126 + 1 = 127
127 + -1n + -6n2 = -1 + n + 6n2 + 1 + -1n + -6n2

Reorder the terms:
127 + -1n + -6n2 = -1 + 1 + n + -1n + 6n2 + -6n2

Combine like terms: -1 + 1 = 0
127 + -1n + -6n2 = 0 + n + -1n + 6n2 + -6n2
127 + -1n + -6n2 = n + -1n + 6n2 + -6n2

Combine like terms: n + -1n = 0
127 + -1n + -6n2 = 0 + 6n2 + -6n2
127 + -1n + -6n2 = 6n2 + -6n2

Combine like terms: 6n2 + -6n2 = 0
127 + -1n + -6n2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-21.16666667 + 0.1666666667n + n2 = 0

Move the constant term to the right:

Add '21.16666667' to each side of the equation.
-21.16666667 + 0.1666666667n + 21.16666667 + n2 = 0 + 21.16666667

Reorder the terms:
-21.16666667 + 21.16666667 + 0.1666666667n + n2 = 0 + 21.16666667

Combine like terms: -21.16666667 + 21.16666667 = 0.00000000
0.00000000 + 0.1666666667n + n2 = 0 + 21.16666667
0.1666666667n + n2 = 0 + 21.16666667

Combine like terms: 0 + 21.16666667 = 21.16666667
0.1666666667n + n2 = 21.16666667

The n term is 0.1666666667n.  Take half its coefficient (0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
0.1666666667n + 0.006944444447 + n2 = 21.16666667 + 0.006944444447

Reorder the terms:
0.006944444447 + 0.1666666667n + n2 = 21.16666667 + 0.006944444447

Combine like terms: 21.16666667 + 0.006944444447 = 21.173611114447
0.006944444447 + 0.1666666667n + n2 = 21.173611114447

Factor a perfect square on the left side:
(n + 0.08333333335)(n + 0.08333333335) = 21.173611114447

Calculate the square root of the right side: 4.601479231

Break this problem into two subproblems by setting 
(n + 0.08333333335) equal to 4.601479231 and -4.601479231.

Subproblem 1

n + 0.08333333335 = 4.601479231 Simplifying n + 0.08333333335 = 4.601479231 Reorder the terms: 0.08333333335 + n = 4.601479231 Solving 0.08333333335 + n = 4.601479231 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + n = 4.601479231 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + n = 4.601479231 + -0.08333333335 n = 4.601479231 + -0.08333333335 Combine like terms: 4.601479231 + -0.08333333335 = 4.51814589765 n = 4.51814589765 Simplifying n = 4.51814589765

Subproblem 2

n + 0.08333333335 = -4.601479231 Simplifying n + 0.08333333335 = -4.601479231 Reorder the terms: 0.08333333335 + n = -4.601479231 Solving 0.08333333335 + n = -4.601479231 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + n = -4.601479231 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + n = -4.601479231 + -0.08333333335 n = -4.601479231 + -0.08333333335 Combine like terms: -4.601479231 + -0.08333333335 = -4.68481256435 n = -4.68481256435 Simplifying n = -4.68481256435

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.51814589765, -4.68481256435}

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