31000=2.6n^2+7n-2

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Solution for 31000=2.6n^2+7n-2 equation:


Simplifying
31000 = 2.6n2 + 7n + -2

Reorder the terms:
31000 = -2 + 7n + 2.6n2

Solving
31000 = -2 + 7n + 2.6n2

Solving for variable 'n'.

Combine like terms: 31000 + 2 = 31002
31002 + -7n + -2.6n2 = -2 + 7n + 2.6n2 + 2 + -7n + -2.6n2

Reorder the terms:
31002 + -7n + -2.6n2 = -2 + 2 + 7n + -7n + 2.6n2 + -2.6n2

Combine like terms: -2 + 2 = 0
31002 + -7n + -2.6n2 = 0 + 7n + -7n + 2.6n2 + -2.6n2
31002 + -7n + -2.6n2 = 7n + -7n + 2.6n2 + -2.6n2

Combine like terms: 7n + -7n = 0
31002 + -7n + -2.6n2 = 0 + 2.6n2 + -2.6n2
31002 + -7n + -2.6n2 = 2.6n2 + -2.6n2

Combine like terms: 2.6n2 + -2.6n2 = 0.0
31002 + -7n + -2.6n2 = 0.0

Begin completing the square.  Divide all terms by
-2.6 the coefficient of the squared term: 

Divide each side by '-2.6'.
-11923.84615 + 2.692307692n + n2 = 0

Move the constant term to the right:

Add '11923.84615' to each side of the equation.
-11923.84615 + 2.692307692n + 11923.84615 + n2 = 0 + 11923.84615

Reorder the terms:
-11923.84615 + 11923.84615 + 2.692307692n + n2 = 0 + 11923.84615

Combine like terms: -11923.84615 + 11923.84615 = 0.00000
0.00000 + 2.692307692n + n2 = 0 + 11923.84615
2.692307692n + n2 = 0 + 11923.84615

Combine like terms: 0 + 11923.84615 = 11923.84615
2.692307692n + n2 = 11923.84615

The n term is 2.692307692n.  Take half its coefficient (1.346153846).
Square it (1.812130177) and add it to both sides.

Add '1.812130177' to each side of the equation.
2.692307692n + 1.812130177 + n2 = 11923.84615 + 1.812130177

Reorder the terms:
1.812130177 + 2.692307692n + n2 = 11923.84615 + 1.812130177

Combine like terms: 11923.84615 + 1.812130177 = 11925.658280177
1.812130177 + 2.692307692n + n2 = 11925.658280177

Factor a perfect square on the left side:
(n + 1.346153846)(n + 1.346153846) = 11925.658280177

Calculate the square root of the right side: 109.204662355

Break this problem into two subproblems by setting 
(n + 1.346153846) equal to 109.204662355 and -109.204662355.

Subproblem 1

n + 1.346153846 = 109.204662355 Simplifying n + 1.346153846 = 109.204662355 Reorder the terms: 1.346153846 + n = 109.204662355 Solving 1.346153846 + n = 109.204662355 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.346153846' to each side of the equation. 1.346153846 + -1.346153846 + n = 109.204662355 + -1.346153846 Combine like terms: 1.346153846 + -1.346153846 = 0.000000000 0.000000000 + n = 109.204662355 + -1.346153846 n = 109.204662355 + -1.346153846 Combine like terms: 109.204662355 + -1.346153846 = 107.858508509 n = 107.858508509 Simplifying n = 107.858508509

Subproblem 2

n + 1.346153846 = -109.204662355 Simplifying n + 1.346153846 = -109.204662355 Reorder the terms: 1.346153846 + n = -109.204662355 Solving 1.346153846 + n = -109.204662355 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.346153846' to each side of the equation. 1.346153846 + -1.346153846 + n = -109.204662355 + -1.346153846 Combine like terms: 1.346153846 + -1.346153846 = 0.000000000 0.000000000 + n = -109.204662355 + -1.346153846 n = -109.204662355 + -1.346153846 Combine like terms: -109.204662355 + -1.346153846 = -110.550816201 n = -110.550816201 Simplifying n = -110.550816201

Solution

The solution to the problem is based on the solutions from the subproblems. n = {107.858508509, -110.550816201}

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