n^2-6n=1

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Solution for n^2-6n=1 equation:


Simplifying
n2 + -6n = 1

Reorder the terms:
-6n + n2 = 1

Solving
-6n + n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -6n + n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -6n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -6n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -6n + n2 = 0 + 1
-6n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-6n + n2 = 1

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = 1 + 9

Reorder the terms:
9 + -6n + n2 = 1 + 9

Combine like terms: 1 + 9 = 10
9 + -6n + n2 = 10

Factor a perfect square on the left side:
(n + -3)(n + -3) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(n + -3) equal to 3.16227766 and -3.16227766.

Subproblem 1

n + -3 = 3.16227766 Simplifying n + -3 = 3.16227766 Reorder the terms: -3 + n = 3.16227766 Solving -3 + n = 3.16227766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 3.16227766 + 3 Combine like terms: -3 + 3 = 0 0 + n = 3.16227766 + 3 n = 3.16227766 + 3 Combine like terms: 3.16227766 + 3 = 6.16227766 n = 6.16227766 Simplifying n = 6.16227766

Subproblem 2

n + -3 = -3.16227766 Simplifying n + -3 = -3.16227766 Reorder the terms: -3 + n = -3.16227766 Solving -3 + n = -3.16227766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -3.16227766 + 3 Combine like terms: -3 + 3 = 0 0 + n = -3.16227766 + 3 n = -3.16227766 + 3 Combine like terms: -3.16227766 + 3 = -0.16227766 n = -0.16227766 Simplifying n = -0.16227766

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.16227766, -0.16227766}

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