7n^2-n=15

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Solution for 7n^2-n=15 equation:


Simplifying
7n2 + -1n = 15

Reorder the terms:
-1n + 7n2 = 15

Solving
-1n + 7n2 = 15

Solving for variable 'n'.

Reorder the terms:
-15 + -1n + 7n2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + -1n + 7n2 = 0

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

Divide each side by '7'.
-2.142857143 + -0.1428571429n + n2 = 0

Move the constant term to the right:

Add '2.142857143' to each side of the equation.
-2.142857143 + -0.1428571429n + 2.142857143 + n2 = 0 + 2.142857143

Reorder the terms:
-2.142857143 + 2.142857143 + -0.1428571429n + n2 = 0 + 2.142857143

Combine like terms: -2.142857143 + 2.142857143 = 0.000000000
0.000000000 + -0.1428571429n + n2 = 0 + 2.142857143
-0.1428571429n + n2 = 0 + 2.142857143

Combine like terms: 0 + 2.142857143 = 2.142857143
-0.1428571429n + n2 = 2.142857143

The n term is -0.1428571429n.  Take half its coefficient (-0.07142857145).
Square it (0.005102040819) and add it to both sides.

Add '0.005102040819' to each side of the equation.
-0.1428571429n + 0.005102040819 + n2 = 2.142857143 + 0.005102040819

Reorder the terms:
0.005102040819 + -0.1428571429n + n2 = 2.142857143 + 0.005102040819

Combine like terms: 2.142857143 + 0.005102040819 = 2.147959183819
0.005102040819 + -0.1428571429n + n2 = 2.147959183819

Factor a perfect square on the left side:
(n + -0.07142857145)(n + -0.07142857145) = 2.147959183819

Calculate the square root of the right side: 1.465591752

Break this problem into two subproblems by setting 
(n + -0.07142857145) equal to 1.465591752 and -1.465591752.

Subproblem 1

n + -0.07142857145 = 1.465591752 Simplifying n + -0.07142857145 = 1.465591752 Reorder the terms: -0.07142857145 + n = 1.465591752 Solving -0.07142857145 + n = 1.465591752 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.07142857145' to each side of the equation. -0.07142857145 + 0.07142857145 + n = 1.465591752 + 0.07142857145 Combine like terms: -0.07142857145 + 0.07142857145 = 0.00000000000 0.00000000000 + n = 1.465591752 + 0.07142857145 n = 1.465591752 + 0.07142857145 Combine like terms: 1.465591752 + 0.07142857145 = 1.53702032345 n = 1.53702032345 Simplifying n = 1.53702032345

Subproblem 2

n + -0.07142857145 = -1.465591752 Simplifying n + -0.07142857145 = -1.465591752 Reorder the terms: -0.07142857145 + n = -1.465591752 Solving -0.07142857145 + n = -1.465591752 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.07142857145' to each side of the equation. -0.07142857145 + 0.07142857145 + n = -1.465591752 + 0.07142857145 Combine like terms: -0.07142857145 + 0.07142857145 = 0.00000000000 0.00000000000 + n = -1.465591752 + 0.07142857145 n = -1.465591752 + 0.07142857145 Combine like terms: -1.465591752 + 0.07142857145 = -1.39416318055 n = -1.39416318055 Simplifying n = -1.39416318055

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.53702032345, -1.39416318055}

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