n^2=12n-7

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Solution for n^2=12n-7 equation:


Simplifying
n2 = 12n + -7

Reorder the terms:
n2 = -7 + 12n

Solving
n2 = -7 + 12n

Solving for variable 'n'.

Reorder the terms:
7 + -12n + n2 = -7 + 12n + 7 + -12n

Reorder the terms:
7 + -12n + n2 = -7 + 7 + 12n + -12n

Combine like terms: -7 + 7 = 0
7 + -12n + n2 = 0 + 12n + -12n
7 + -12n + n2 = 12n + -12n

Combine like terms: 12n + -12n = 0
7 + -12n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -12n + -7 + n2 = 0 + -7

Reorder the terms:
7 + -7 + -12n + n2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -12n + n2 = 0 + -7
-12n + n2 = 0 + -7

Combine like terms: 0 + -7 = -7
-12n + n2 = -7

The n term is -12n.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12n + 36 + n2 = -7 + 36

Reorder the terms:
36 + -12n + n2 = -7 + 36

Combine like terms: -7 + 36 = 29
36 + -12n + n2 = 29

Factor a perfect square on the left side:
(n + -6)(n + -6) = 29

Calculate the square root of the right side: 5.385164807

Break this problem into two subproblems by setting 
(n + -6) equal to 5.385164807 and -5.385164807.

Subproblem 1

n + -6 = 5.385164807 Simplifying n + -6 = 5.385164807 Reorder the terms: -6 + n = 5.385164807 Solving -6 + n = 5.385164807 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 5.385164807 + 6 Combine like terms: -6 + 6 = 0 0 + n = 5.385164807 + 6 n = 5.385164807 + 6 Combine like terms: 5.385164807 + 6 = 11.385164807 n = 11.385164807 Simplifying n = 11.385164807

Subproblem 2

n + -6 = -5.385164807 Simplifying n + -6 = -5.385164807 Reorder the terms: -6 + n = -5.385164807 Solving -6 + n = -5.385164807 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -5.385164807 + 6 Combine like terms: -6 + 6 = 0 0 + n = -5.385164807 + 6 n = -5.385164807 + 6 Combine like terms: -5.385164807 + 6 = 0.614835193 n = 0.614835193 Simplifying n = 0.614835193

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.385164807, 0.614835193}

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