2n(7n+8)=n+8

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Solution for 2n(7n+8)=n+8 equation:


Simplifying
2n(7n + 8) = n + 8

Reorder the terms:
2n(8 + 7n) = n + 8
(8 * 2n + 7n * 2n) = n + 8
(16n + 14n2) = n + 8

Reorder the terms:
16n + 14n2 = 8 + n

Solving
16n + 14n2 = 8 + n

Solving for variable 'n'.

Reorder the terms:
-8 + 16n + -1n + 14n2 = 8 + n + -8 + -1n

Combine like terms: 16n + -1n = 15n
-8 + 15n + 14n2 = 8 + n + -8 + -1n

Reorder the terms:
-8 + 15n + 14n2 = 8 + -8 + n + -1n

Combine like terms: 8 + -8 = 0
-8 + 15n + 14n2 = 0 + n + -1n
-8 + 15n + 14n2 = n + -1n

Combine like terms: n + -1n = 0
-8 + 15n + 14n2 = 0

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

Divide each side by '14'.
-0.5714285714 + 1.071428571n + n2 = 0

Move the constant term to the right:

Add '0.5714285714' to each side of the equation.
-0.5714285714 + 1.071428571n + 0.5714285714 + n2 = 0 + 0.5714285714

Reorder the terms:
-0.5714285714 + 0.5714285714 + 1.071428571n + n2 = 0 + 0.5714285714

Combine like terms: -0.5714285714 + 0.5714285714 = 0.0000000000
0.0000000000 + 1.071428571n + n2 = 0 + 0.5714285714
1.071428571n + n2 = 0 + 0.5714285714

Combine like terms: 0 + 0.5714285714 = 0.5714285714
1.071428571n + n2 = 0.5714285714

The n term is 1.071428571n.  Take half its coefficient (0.5357142855).
Square it (0.2869897957) and add it to both sides.

Add '0.2869897957' to each side of the equation.
1.071428571n + 0.2869897957 + n2 = 0.5714285714 + 0.2869897957

Reorder the terms:
0.2869897957 + 1.071428571n + n2 = 0.5714285714 + 0.2869897957

Combine like terms: 0.5714285714 + 0.2869897957 = 0.8584183671
0.2869897957 + 1.071428571n + n2 = 0.8584183671

Factor a perfect square on the left side:
(n + 0.5357142855)(n + 0.5357142855) = 0.8584183671

Calculate the square root of the right side: 0.926508698

Break this problem into two subproblems by setting 
(n + 0.5357142855) equal to 0.926508698 and -0.926508698.

Subproblem 1

n + 0.5357142855 = 0.926508698 Simplifying n + 0.5357142855 = 0.926508698 Reorder the terms: 0.5357142855 + n = 0.926508698 Solving 0.5357142855 + n = 0.926508698 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5357142855' to each side of the equation. 0.5357142855 + -0.5357142855 + n = 0.926508698 + -0.5357142855 Combine like terms: 0.5357142855 + -0.5357142855 = 0.0000000000 0.0000000000 + n = 0.926508698 + -0.5357142855 n = 0.926508698 + -0.5357142855 Combine like terms: 0.926508698 + -0.5357142855 = 0.3907944125 n = 0.3907944125 Simplifying n = 0.3907944125

Subproblem 2

n + 0.5357142855 = -0.926508698 Simplifying n + 0.5357142855 = -0.926508698 Reorder the terms: 0.5357142855 + n = -0.926508698 Solving 0.5357142855 + n = -0.926508698 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5357142855' to each side of the equation. 0.5357142855 + -0.5357142855 + n = -0.926508698 + -0.5357142855 Combine like terms: 0.5357142855 + -0.5357142855 = 0.0000000000 0.0000000000 + n = -0.926508698 + -0.5357142855 n = -0.926508698 + -0.5357142855 Combine like terms: -0.926508698 + -0.5357142855 = -1.4622229835 n = -1.4622229835 Simplifying n = -1.4622229835

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.3907944125, -1.4622229835}

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