n^2+1=14n

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Solution for n^2+1=14n equation:


Simplifying
n2 + 1 = 14n

Reorder the terms:
1 + n2 = 14n

Solving
1 + n2 = 14n

Solving for variable 'n'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -14n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -14n + n2 = 0 + -1

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

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

The n term is -14n.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14n + 49 + n2 = -1 + 49

Reorder the terms:
49 + -14n + n2 = -1 + 49

Combine like terms: -1 + 49 = 48
49 + -14n + n2 = 48

Factor a perfect square on the left side:
(n + -7)(n + -7) = 48

Calculate the square root of the right side: 6.92820323

Break this problem into two subproblems by setting 
(n + -7) equal to 6.92820323 and -6.92820323.

Subproblem 1

n + -7 = 6.92820323 Simplifying n + -7 = 6.92820323 Reorder the terms: -7 + n = 6.92820323 Solving -7 + n = 6.92820323 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + n = 6.92820323 + 7 Combine like terms: -7 + 7 = 0 0 + n = 6.92820323 + 7 n = 6.92820323 + 7 Combine like terms: 6.92820323 + 7 = 13.92820323 n = 13.92820323 Simplifying n = 13.92820323

Subproblem 2

n + -7 = -6.92820323 Simplifying n + -7 = -6.92820323 Reorder the terms: -7 + n = -6.92820323 Solving -7 + n = -6.92820323 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + n = -6.92820323 + 7 Combine like terms: -7 + 7 = 0 0 + n = -6.92820323 + 7 n = -6.92820323 + 7 Combine like terms: -6.92820323 + 7 = 0.07179677 n = 0.07179677 Simplifying n = 0.07179677

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.92820323, 0.07179677}

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