2n^2+20n-86=7

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Solution for 2n^2+20n-86=7 equation:


Simplifying
2n2 + 20n + -86 = 7

Reorder the terms:
-86 + 20n + 2n2 = 7

Solving
-86 + 20n + 2n2 = 7

Solving for variable 'n'.

Reorder the terms:
-86 + -7 + 20n + 2n2 = 7 + -7

Combine like terms: -86 + -7 = -93
-93 + 20n + 2n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-93 + 20n + 2n2 = 0

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

Divide each side by '2'.
-46.5 + 10n + n2 = 0

Move the constant term to the right:

Add '46.5' to each side of the equation.
-46.5 + 10n + 46.5 + n2 = 0 + 46.5

Reorder the terms:
-46.5 + 46.5 + 10n + n2 = 0 + 46.5

Combine like terms: -46.5 + 46.5 = 0.0
0.0 + 10n + n2 = 0 + 46.5
10n + n2 = 0 + 46.5

Combine like terms: 0 + 46.5 = 46.5
10n + n2 = 46.5

The n term is 10n.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10n + 25 + n2 = 46.5 + 25

Reorder the terms:
25 + 10n + n2 = 46.5 + 25

Combine like terms: 46.5 + 25 = 71.5
25 + 10n + n2 = 71.5

Factor a perfect square on the left side:
(n + 5)(n + 5) = 71.5

Calculate the square root of the right side: 8.455767263

Break this problem into two subproblems by setting 
(n + 5) equal to 8.455767263 and -8.455767263.

Subproblem 1

n + 5 = 8.455767263 Simplifying n + 5 = 8.455767263 Reorder the terms: 5 + n = 8.455767263 Solving 5 + n = 8.455767263 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 8.455767263 + -5 Combine like terms: 5 + -5 = 0 0 + n = 8.455767263 + -5 n = 8.455767263 + -5 Combine like terms: 8.455767263 + -5 = 3.455767263 n = 3.455767263 Simplifying n = 3.455767263

Subproblem 2

n + 5 = -8.455767263 Simplifying n + 5 = -8.455767263 Reorder the terms: 5 + n = -8.455767263 Solving 5 + n = -8.455767263 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -8.455767263 + -5 Combine like terms: 5 + -5 = 0 0 + n = -8.455767263 + -5 n = -8.455767263 + -5 Combine like terms: -8.455767263 + -5 = -13.455767263 n = -13.455767263 Simplifying n = -13.455767263

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.455767263, -13.455767263}

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