n^2-13n=1

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Solution for n^2-13n=1 equation:


Simplifying
n2 + -13n = 1

Reorder the terms:
-13n + n2 = 1

Solving
-13n + n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -13n + n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -13n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -13n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -13n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -13n + n2 = 0 + 1
-13n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-13n + n2 = 1

The n term is -13n.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13n + 42.25 + n2 = 1 + 42.25

Reorder the terms:
42.25 + -13n + n2 = 1 + 42.25

Combine like terms: 1 + 42.25 = 43.25
42.25 + -13n + n2 = 43.25

Factor a perfect square on the left side:
(n + -6.5)(n + -6.5) = 43.25

Calculate the square root of the right side: 6.576473219

Break this problem into two subproblems by setting 
(n + -6.5) equal to 6.576473219 and -6.576473219.

Subproblem 1

n + -6.5 = 6.576473219 Simplifying n + -6.5 = 6.576473219 Reorder the terms: -6.5 + n = 6.576473219 Solving -6.5 + n = 6.576473219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + n = 6.576473219 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = 6.576473219 + 6.5 n = 6.576473219 + 6.5 Combine like terms: 6.576473219 + 6.5 = 13.076473219 n = 13.076473219 Simplifying n = 13.076473219

Subproblem 2

n + -6.5 = -6.576473219 Simplifying n + -6.5 = -6.576473219 Reorder the terms: -6.5 + n = -6.576473219 Solving -6.5 + n = -6.576473219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + n = -6.576473219 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = -6.576473219 + 6.5 n = -6.576473219 + 6.5 Combine like terms: -6.576473219 + 6.5 = -0.076473219 n = -0.076473219 Simplifying n = -0.076473219

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.076473219, -0.076473219}

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