0=n^2+n-1600

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Solution for 0=n^2+n-1600 equation:


Simplifying
0 = n2 + n + -1600

Reorder the terms:
0 = -1600 + n + n2

Solving
0 = -1600 + n + n2

Solving for variable 'n'.

Combine like terms: 0 + 1600 = 1600
1600 + -1n + -1n2 = -1600 + n + n2 + 1600 + -1n + -1n2

Reorder the terms:
1600 + -1n + -1n2 = -1600 + 1600 + n + -1n + n2 + -1n2

Combine like terms: -1600 + 1600 = 0
1600 + -1n + -1n2 = 0 + n + -1n + n2 + -1n2
1600 + -1n + -1n2 = n + -1n + n2 + -1n2

Combine like terms: n + -1n = 0
1600 + -1n + -1n2 = 0 + n2 + -1n2
1600 + -1n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
1600 + -1n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1600 + n + n2 = 0

Move the constant term to the right:

Add '1600' to each side of the equation.
-1600 + n + 1600 + n2 = 0 + 1600

Reorder the terms:
-1600 + 1600 + n + n2 = 0 + 1600

Combine like terms: -1600 + 1600 = 0
0 + n + n2 = 0 + 1600
n + n2 = 0 + 1600

Combine like terms: 0 + 1600 = 1600
n + n2 = 1600

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.
 + 0.25 + n2 = 1600 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 1600 + 0.25

Combine like terms: 1600 + 0.25 = 1600.25
1.25 + n2 = 1600.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 1600.25

Calculate the square root of the right side: 40.003124878

Break this problem into two subproblems by setting 
(n + 0.5) equal to 40.003124878 and -40.003124878.

Subproblem 1

n + 0.5 = 40.003124878 Simplifying n + 0.5 = 40.003124878 Reorder the terms: 0.5 + n = 40.003124878 Solving 0.5 + n = 40.003124878 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 = 40.003124878 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 40.003124878 + -0.5 n = 40.003124878 + -0.5 Combine like terms: 40.003124878 + -0.5 = 39.503124878 n = 39.503124878 Simplifying n = 39.503124878

Subproblem 2

n + 0.5 = -40.003124878 Simplifying n + 0.5 = -40.003124878 Reorder the terms: 0.5 + n = -40.003124878 Solving 0.5 + n = -40.003124878 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 = -40.003124878 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -40.003124878 + -0.5 n = -40.003124878 + -0.5 Combine like terms: -40.003124878 + -0.5 = -40.503124878 n = -40.503124878 Simplifying n = -40.503124878

Solution

The solution to the problem is based on the solutions from the subproblems. n = {39.503124878, -40.503124878}

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