n^2+5n-120=0

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Solution for n^2+5n-120=0 equation:


Simplifying
n2 + 5n + -120 = 0

Reorder the terms:
-120 + 5n + n2 = 0

Solving
-120 + 5n + 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 + 5n + 120 + n2 = 0 + 120

Reorder the terms:
-120 + 120 + 5n + n2 = 0 + 120

Combine like terms: -120 + 120 = 0
0 + 5n + n2 = 0 + 120
5n + n2 = 0 + 120

Combine like terms: 0 + 120 = 120
5n + n2 = 120

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = 120 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 120 + 6.25

Combine like terms: 120 + 6.25 = 126.25
6.25 + 5n + n2 = 126.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 126.25

Calculate the square root of the right side: 11.236102527

Break this problem into two subproblems by setting 
(n + 2.5) equal to 11.236102527 and -11.236102527.

Subproblem 1

n + 2.5 = 11.236102527 Simplifying n + 2.5 = 11.236102527 Reorder the terms: 2.5 + n = 11.236102527 Solving 2.5 + n = 11.236102527 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 11.236102527 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 11.236102527 + -2.5 n = 11.236102527 + -2.5 Combine like terms: 11.236102527 + -2.5 = 8.736102527 n = 8.736102527 Simplifying n = 8.736102527

Subproblem 2

n + 2.5 = -11.236102527 Simplifying n + 2.5 = -11.236102527 Reorder the terms: 2.5 + n = -11.236102527 Solving 2.5 + n = -11.236102527 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -11.236102527 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -11.236102527 + -2.5 n = -11.236102527 + -2.5 Combine like terms: -11.236102527 + -2.5 = -13.736102527 n = -13.736102527 Simplifying n = -13.736102527

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.736102527, -13.736102527}

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