n^2-n-105=0

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Solution for n^2-n-105=0 equation:


Simplifying
n2 + -1n + -105 = 0

Reorder the terms:
-105 + -1n + n2 = 0

Solving
-105 + -1n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '105' to each side of the equation.
-105 + -1n + 105 + n2 = 0 + 105

Reorder the terms:
-105 + 105 + -1n + n2 = 0 + 105

Combine like terms: -105 + 105 = 0
0 + -1n + n2 = 0 + 105
-1n + n2 = 0 + 105

Combine like terms: 0 + 105 = 105
-1n + n2 = 105

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 = 105 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 105 + 0.25

Combine like terms: 105 + 0.25 = 105.25
0.25 + -1n + n2 = 105.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 105.25

Calculate the square root of the right side: 10.259142264

Break this problem into two subproblems by setting 
(n + -0.5) equal to 10.259142264 and -10.259142264.

Subproblem 1

n + -0.5 = 10.259142264 Simplifying n + -0.5 = 10.259142264 Reorder the terms: -0.5 + n = 10.259142264 Solving -0.5 + n = 10.259142264 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 = 10.259142264 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 10.259142264 + 0.5 n = 10.259142264 + 0.5 Combine like terms: 10.259142264 + 0.5 = 10.759142264 n = 10.759142264 Simplifying n = 10.759142264

Subproblem 2

n + -0.5 = -10.259142264 Simplifying n + -0.5 = -10.259142264 Reorder the terms: -0.5 + n = -10.259142264 Solving -0.5 + n = -10.259142264 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 = -10.259142264 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -10.259142264 + 0.5 n = -10.259142264 + 0.5 Combine like terms: -10.259142264 + 0.5 = -9.759142264 n = -9.759142264 Simplifying n = -9.759142264

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.759142264, -9.759142264}

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