n^2-n-56=1

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Solution for n^2-n-56=1 equation:


Simplifying
n2 + -1n + -56 = 1

Reorder the terms:
-56 + -1n + n2 = 1

Solving
-56 + -1n + n2 = 1

Solving for variable 'n'.

Reorder the terms:
-56 + -1 + -1n + n2 = 1 + -1

Combine like terms: -56 + -1 = -57
-57 + -1n + n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-57 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '57' to each side of the equation.
-57 + -1n + 57 + n2 = 0 + 57

Reorder the terms:
-57 + 57 + -1n + n2 = 0 + 57

Combine like terms: -57 + 57 = 0
0 + -1n + n2 = 0 + 57
-1n + n2 = 0 + 57

Combine like terms: 0 + 57 = 57
-1n + n2 = 57

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 57 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 57 + 0.25

Combine like terms: 57 + 0.25 = 57.25
0.25 + -1n + n2 = 57.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 57.25

Calculate the square root of the right side: 7.566372975

Break this problem into two subproblems by setting 
(n + -0.5) equal to 7.566372975 and -7.566372975.

Subproblem 1

n + -0.5 = 7.566372975 Simplifying n + -0.5 = 7.566372975 Reorder the terms: -0.5 + n = 7.566372975 Solving -0.5 + n = 7.566372975 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 7.566372975 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 7.566372975 + 0.5 n = 7.566372975 + 0.5 Combine like terms: 7.566372975 + 0.5 = 8.066372975 n = 8.066372975 Simplifying n = 8.066372975

Subproblem 2

n + -0.5 = -7.566372975 Simplifying n + -0.5 = -7.566372975 Reorder the terms: -0.5 + n = -7.566372975 Solving -0.5 + n = -7.566372975 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -7.566372975 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -7.566372975 + 0.5 n = -7.566372975 + 0.5 Combine like terms: -7.566372975 + 0.5 = -7.066372975 n = -7.066372975 Simplifying n = -7.066372975

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.066372975, -7.066372975}

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