7n^2+2n-37=0

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Solution for 7n^2+2n-37=0 equation:


Simplifying
7n2 + 2n + -37 = 0

Reorder the terms:
-37 + 2n + 7n2 = 0

Solving
-37 + 2n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
-5.285714286 + 0.2857142857n + n2 = 0

Move the constant term to the right:

Add '5.285714286' to each side of the equation.
-5.285714286 + 0.2857142857n + 5.285714286 + n2 = 0 + 5.285714286

Reorder the terms:
-5.285714286 + 5.285714286 + 0.2857142857n + n2 = 0 + 5.285714286

Combine like terms: -5.285714286 + 5.285714286 = 0.000000000
0.000000000 + 0.2857142857n + n2 = 0 + 5.285714286
0.2857142857n + n2 = 0 + 5.285714286

Combine like terms: 0 + 5.285714286 = 5.285714286
0.2857142857n + n2 = 5.285714286

The n term is 0.2857142857n.  Take half its coefficient (0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
0.2857142857n + 0.02040816328 + n2 = 5.285714286 + 0.02040816328

Reorder the terms:
0.02040816328 + 0.2857142857n + n2 = 5.285714286 + 0.02040816328

Combine like terms: 5.285714286 + 0.02040816328 = 5.30612244928
0.02040816328 + 0.2857142857n + n2 = 5.30612244928

Factor a perfect square on the left side:
(n + 0.1428571429)(n + 0.1428571429) = 5.30612244928

Calculate the square root of the right side: 2.303502214

Break this problem into two subproblems by setting 
(n + 0.1428571429) equal to 2.303502214 and -2.303502214.

Subproblem 1

n + 0.1428571429 = 2.303502214 Simplifying n + 0.1428571429 = 2.303502214 Reorder the terms: 0.1428571429 + n = 2.303502214 Solving 0.1428571429 + n = 2.303502214 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + n = 2.303502214 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + n = 2.303502214 + -0.1428571429 n = 2.303502214 + -0.1428571429 Combine like terms: 2.303502214 + -0.1428571429 = 2.1606450711 n = 2.1606450711 Simplifying n = 2.1606450711

Subproblem 2

n + 0.1428571429 = -2.303502214 Simplifying n + 0.1428571429 = -2.303502214 Reorder the terms: 0.1428571429 + n = -2.303502214 Solving 0.1428571429 + n = -2.303502214 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + n = -2.303502214 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + n = -2.303502214 + -0.1428571429 n = -2.303502214 + -0.1428571429 Combine like terms: -2.303502214 + -0.1428571429 = -2.4463593569 n = -2.4463593569 Simplifying n = -2.4463593569

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.1606450711, -2.4463593569}

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