n^2+n-120=0

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Solution for n^2+n-120=0 equation:


Simplifying
n2 + n + -120 = 0

Reorder the terms:
-120 + n + n2 = 0

Solving
-120 + n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '120' to each side of the equation.
-120 + n + 120 + n2 = 0 + 120

Reorder the terms:
-120 + 120 + n + n2 = 0 + 120

Combine like terms: -120 + 120 = 0
0 + n + n2 = 0 + 120
n + n2 = 0 + 120

Combine like terms: 0 + 120 = 120
n + n2 = 120

The n term is n.  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.
n + 0.25 + n2 = 120 + 0.25

Reorder the terms:
0.25 + n + n2 = 120 + 0.25

Combine like terms: 120 + 0.25 = 120.25
0.25 + n + n2 = 120.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 120.25

Calculate the square root of the right side: 10.9658561

Break this problem into two subproblems by setting 
(n + 0.5) equal to 10.9658561 and -10.9658561.

Subproblem 1

n + 0.5 = 10.9658561 Simplifying n + 0.5 = 10.9658561 Reorder the terms: 0.5 + n = 10.9658561 Solving 0.5 + n = 10.9658561 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 = 10.9658561 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 10.9658561 + -0.5 n = 10.9658561 + -0.5 Combine like terms: 10.9658561 + -0.5 = 10.4658561 n = 10.4658561 Simplifying n = 10.4658561

Subproblem 2

n + 0.5 = -10.9658561 Simplifying n + 0.5 = -10.9658561 Reorder the terms: 0.5 + n = -10.9658561 Solving 0.5 + n = -10.9658561 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 = -10.9658561 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -10.9658561 + -0.5 n = -10.9658561 + -0.5 Combine like terms: -10.9658561 + -0.5 = -11.4658561 n = -11.4658561 Simplifying n = -11.4658561

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.4658561, -11.4658561}

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