n^2+n=4200

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Solution for n^2+n=4200 equation:


Simplifying
n2 + n = 4200

Reorder the terms:
n + n2 = 4200

Solving
n + n2 = 4200

Solving for variable 'n'.

Reorder the terms:
-4200 + n + n2 = 4200 + -4200

Combine like terms: 4200 + -4200 = 0
-4200 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4200' to each side of the equation.
-4200 + n + 4200 + n2 = 0 + 4200

Reorder the terms:
-4200 + 4200 + n + n2 = 0 + 4200

Combine like terms: -4200 + 4200 = 0
0 + n + n2 = 0 + 4200
n + n2 = 0 + 4200

Combine like terms: 0 + 4200 = 4200
n + n2 = 4200

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 = 4200 + 0.25

Reorder the terms:
0.25 + n + n2 = 4200 + 0.25

Combine like terms: 4200 + 0.25 = 4200.25
0.25 + n + n2 = 4200.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 4200.25

Calculate the square root of the right side: 64.809335747

Break this problem into two subproblems by setting 
(n + 0.5) equal to 64.809335747 and -64.809335747.

Subproblem 1

n + 0.5 = 64.809335747 Simplifying n + 0.5 = 64.809335747 Reorder the terms: 0.5 + n = 64.809335747 Solving 0.5 + n = 64.809335747 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 = 64.809335747 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 64.809335747 + -0.5 n = 64.809335747 + -0.5 Combine like terms: 64.809335747 + -0.5 = 64.309335747 n = 64.309335747 Simplifying n = 64.309335747

Subproblem 2

n + 0.5 = -64.809335747 Simplifying n + 0.5 = -64.809335747 Reorder the terms: 0.5 + n = -64.809335747 Solving 0.5 + n = -64.809335747 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 = -64.809335747 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -64.809335747 + -0.5 n = -64.809335747 + -0.5 Combine like terms: -64.809335747 + -0.5 = -65.309335747 n = -65.309335747 Simplifying n = -65.309335747

Solution

The solution to the problem is based on the solutions from the subproblems. n = {64.309335747, -65.309335747}

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