7n^2-8n=14

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Solution for 7n^2-8n=14 equation:


Simplifying
7n2 + -8n = 14

Reorder the terms:
-8n + 7n2 = 14

Solving
-8n + 7n2 = 14

Solving for variable 'n'.

Reorder the terms:
-14 + -8n + 7n2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + -8n + 7n2 = 0

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

Divide each side by '7'.
-2 + -1.142857143n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.142857143n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.142857143n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.142857143n + n2 = 0 + 2
-1.142857143n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.142857143n + n2 = 2

The n term is -1.142857143n.  Take half its coefficient (-0.5714285715).
Square it (0.3265306123) and add it to both sides.

Add '0.3265306123' to each side of the equation.
-1.142857143n + 0.3265306123 + n2 = 2 + 0.3265306123

Reorder the terms:
0.3265306123 + -1.142857143n + n2 = 2 + 0.3265306123

Combine like terms: 2 + 0.3265306123 = 2.3265306123
0.3265306123 + -1.142857143n + n2 = 2.3265306123

Factor a perfect square on the left side:
(n + -0.5714285715)(n + -0.5714285715) = 2.3265306123

Calculate the square root of the right side: 1.525296893

Break this problem into two subproblems by setting 
(n + -0.5714285715) equal to 1.525296893 and -1.525296893.

Subproblem 1

n + -0.5714285715 = 1.525296893 Simplifying n + -0.5714285715 = 1.525296893 Reorder the terms: -0.5714285715 + n = 1.525296893 Solving -0.5714285715 + n = 1.525296893 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + n = 1.525296893 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + n = 1.525296893 + 0.5714285715 n = 1.525296893 + 0.5714285715 Combine like terms: 1.525296893 + 0.5714285715 = 2.0967254645 n = 2.0967254645 Simplifying n = 2.0967254645

Subproblem 2

n + -0.5714285715 = -1.525296893 Simplifying n + -0.5714285715 = -1.525296893 Reorder the terms: -0.5714285715 + n = -1.525296893 Solving -0.5714285715 + n = -1.525296893 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + n = -1.525296893 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + n = -1.525296893 + 0.5714285715 n = -1.525296893 + 0.5714285715 Combine like terms: -1.525296893 + 0.5714285715 = -0.9538683215 n = -0.9538683215 Simplifying n = -0.9538683215

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.0967254645, -0.9538683215}

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