n^2-n=54

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Solution for n^2-n=54 equation:


Simplifying
n2 + -1n = 54

Reorder the terms:
-1n + n2 = 54

Solving
-1n + n2 = 54

Solving for variable 'n'.

Reorder the terms:
-54 + -1n + n2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -1n + 54 + n2 = 0 + 54

Reorder the terms:
-54 + 54 + -1n + n2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -1n + n2 = 0 + 54
-1n + n2 = 0 + 54

Combine like terms: 0 + 54 = 54
-1n + n2 = 54

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 = 54 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 54 + 0.25

Combine like terms: 54 + 0.25 = 54.25
0.25 + -1n + n2 = 54.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 54.25

Calculate the square root of the right side: 7.365459931

Break this problem into two subproblems by setting 
(n + -0.5) equal to 7.365459931 and -7.365459931.

Subproblem 1

n + -0.5 = 7.365459931 Simplifying n + -0.5 = 7.365459931 Reorder the terms: -0.5 + n = 7.365459931 Solving -0.5 + n = 7.365459931 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 = 7.365459931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 7.365459931 + 0.5 n = 7.365459931 + 0.5 Combine like terms: 7.365459931 + 0.5 = 7.865459931 n = 7.865459931 Simplifying n = 7.865459931

Subproblem 2

n + -0.5 = -7.365459931 Simplifying n + -0.5 = -7.365459931 Reorder the terms: -0.5 + n = -7.365459931 Solving -0.5 + n = -7.365459931 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 = -7.365459931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -7.365459931 + 0.5 n = -7.365459931 + 0.5 Combine like terms: -7.365459931 + 0.5 = -6.865459931 n = -6.865459931 Simplifying n = -6.865459931

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.865459931, -6.865459931}

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