n^2+3n-4000=0

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Solution for n^2+3n-4000=0 equation:


Simplifying
n2 + 3n + -4000 = 0

Reorder the terms:
-4000 + 3n + n2 = 0

Solving
-4000 + 3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '4000' to each side of the equation.
-4000 + 3n + 4000 + n2 = 0 + 4000

Reorder the terms:
-4000 + 4000 + 3n + n2 = 0 + 4000

Combine like terms: -4000 + 4000 = 0
0 + 3n + n2 = 0 + 4000
3n + n2 = 0 + 4000

Combine like terms: 0 + 4000 = 4000
3n + n2 = 4000

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 4000 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 4000 + 2.25

Combine like terms: 4000 + 2.25 = 4002.25
2.25 + 3n + n2 = 4002.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 4002.25

Calculate the square root of the right side: 63.263338514

Break this problem into two subproblems by setting 
(n + 1.5) equal to 63.263338514 and -63.263338514.

Subproblem 1

n + 1.5 = 63.263338514 Simplifying n + 1.5 = 63.263338514 Reorder the terms: 1.5 + n = 63.263338514 Solving 1.5 + n = 63.263338514 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 63.263338514 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 63.263338514 + -1.5 n = 63.263338514 + -1.5 Combine like terms: 63.263338514 + -1.5 = 61.763338514 n = 61.763338514 Simplifying n = 61.763338514

Subproblem 2

n + 1.5 = -63.263338514 Simplifying n + 1.5 = -63.263338514 Reorder the terms: 1.5 + n = -63.263338514 Solving 1.5 + n = -63.263338514 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -63.263338514 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -63.263338514 + -1.5 n = -63.263338514 + -1.5 Combine like terms: -63.263338514 + -1.5 = -64.763338514 n = -64.763338514 Simplifying n = -64.763338514

Solution

The solution to the problem is based on the solutions from the subproblems. n = {61.763338514, -64.763338514}

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