n-1*3n+1=3n*n-5

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Solution for n-1*3n+1=3n*n-5 equation:


Simplifying
n + -1 * 3n + 1 = 3n * n + -5

Multiply -1 * 3
n + -3n + 1 = 3n * n + -5

Reorder the terms:
1 + n + -3n = 3n * n + -5

Combine like terms: n + -3n = -2n
1 + -2n = 3n * n + -5

Multiply n * n
1 + -2n = 3n2 + -5

Reorder the terms:
1 + -2n = -5 + 3n2

Solving
1 + -2n = -5 + 3n2

Solving for variable 'n'.

Reorder the terms:
1 + 5 + -2n + -3n2 = -5 + 3n2 + 5 + -3n2

Combine like terms: 1 + 5 = 6
6 + -2n + -3n2 = -5 + 3n2 + 5 + -3n2

Reorder the terms:
6 + -2n + -3n2 = -5 + 5 + 3n2 + -3n2

Combine like terms: -5 + 5 = 0
6 + -2n + -3n2 = 0 + 3n2 + -3n2
6 + -2n + -3n2 = 3n2 + -3n2

Combine like terms: 3n2 + -3n2 = 0
6 + -2n + -3n2 = 0

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

Divide each side by '-3'.
-2 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.6666666667n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.6666666667n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.6666666667n + n2 = 0 + 2
0.6666666667n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.6666666667n + n2 = 2

The n term is 0.6666666667n.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667n + 0.1111111112 + n2 = 2 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 2 + 0.1111111112

Combine like terms: 2 + 0.1111111112 = 2.1111111112
0.1111111112 + 0.6666666667n + n2 = 2.1111111112

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 2.1111111112

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 1.452966315 and -1.452966315.

Subproblem 1

n + 0.3333333334 = 1.452966315 Simplifying n + 0.3333333334 = 1.452966315 Reorder the terms: 0.3333333334 + n = 1.452966315 Solving 0.3333333334 + n = 1.452966315 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = 1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 1.452966315 + -0.3333333334 n = 1.452966315 + -0.3333333334 Combine like terms: 1.452966315 + -0.3333333334 = 1.1196329816 n = 1.1196329816 Simplifying n = 1.1196329816

Subproblem 2

n + 0.3333333334 = -1.452966315 Simplifying n + 0.3333333334 = -1.452966315 Reorder the terms: 0.3333333334 + n = -1.452966315 Solving 0.3333333334 + n = -1.452966315 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = -1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -1.452966315 + -0.3333333334 n = -1.452966315 + -0.3333333334 Combine like terms: -1.452966315 + -0.3333333334 = -1.7862996484 n = -1.7862996484 Simplifying n = -1.7862996484

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.1196329816, -1.7862996484}

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