6n^2+7n=7

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Solution for 6n^2+7n=7 equation:


Simplifying
6n2 + 7n = 7

Reorder the terms:
7n + 6n2 = 7

Solving
7n + 6n2 = 7

Solving for variable 'n'.

Reorder the terms:
-7 + 7n + 6n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 7n + 6n2 = 0

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

Divide each side by '6'.
-1.166666667 + 1.166666667n + n2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667n + 1.166666667 + n2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + 1.166666667n + n2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + 1.166666667n + n2 = 0 + 1.166666667
1.166666667n + n2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
1.166666667n + n2 = 1.166666667

The n term is 1.166666667n.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667n + 0.3402777780 + n2 = 1.166666667 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667n + n2 = 1.166666667 + 0.3402777780

Combine like terms: 1.166666667 + 0.3402777780 = 1.506944445
0.3402777780 + 1.166666667n + n2 = 1.506944445

Factor a perfect square on the left side:
(n + 0.5833333335)(n + 0.5833333335) = 1.506944445

Calculate the square root of the right side: 1.227576655

Break this problem into two subproblems by setting 
(n + 0.5833333335) equal to 1.227576655 and -1.227576655.

Subproblem 1

n + 0.5833333335 = 1.227576655 Simplifying n + 0.5833333335 = 1.227576655 Reorder the terms: 0.5833333335 + n = 1.227576655 Solving 0.5833333335 + n = 1.227576655 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + n = 1.227576655 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = 1.227576655 + -0.5833333335 n = 1.227576655 + -0.5833333335 Combine like terms: 1.227576655 + -0.5833333335 = 0.6442433215 n = 0.6442433215 Simplifying n = 0.6442433215

Subproblem 2

n + 0.5833333335 = -1.227576655 Simplifying n + 0.5833333335 = -1.227576655 Reorder the terms: 0.5833333335 + n = -1.227576655 Solving 0.5833333335 + n = -1.227576655 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + n = -1.227576655 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = -1.227576655 + -0.5833333335 n = -1.227576655 + -0.5833333335 Combine like terms: -1.227576655 + -0.5833333335 = -1.8109099885 n = -1.8109099885 Simplifying n = -1.8109099885

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.6442433215, -1.8109099885}

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