500=1000+5n-4n^2

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Solution for 500=1000+5n-4n^2 equation:


Simplifying
500 = 1000 + 5n + -4n2

Solving
500 = 1000 + 5n + -4n2

Solving for variable 'n'.

Combine like terms: 500 + -1000 = -500
-500 + -5n + 4n2 = 1000 + 5n + -4n2 + -1000 + -5n + 4n2

Reorder the terms:
-500 + -5n + 4n2 = 1000 + -1000 + 5n + -5n + -4n2 + 4n2

Combine like terms: 1000 + -1000 = 0
-500 + -5n + 4n2 = 0 + 5n + -5n + -4n2 + 4n2
-500 + -5n + 4n2 = 5n + -5n + -4n2 + 4n2

Combine like terms: 5n + -5n = 0
-500 + -5n + 4n2 = 0 + -4n2 + 4n2
-500 + -5n + 4n2 = -4n2 + 4n2

Combine like terms: -4n2 + 4n2 = 0
-500 + -5n + 4n2 = 0

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

Divide each side by '4'.
-125 + -1.25n + n2 = 0

Move the constant term to the right:

Add '125' to each side of the equation.
-125 + -1.25n + 125 + n2 = 0 + 125

Reorder the terms:
-125 + 125 + -1.25n + n2 = 0 + 125

Combine like terms: -125 + 125 = 0
0 + -1.25n + n2 = 0 + 125
-1.25n + n2 = 0 + 125

Combine like terms: 0 + 125 = 125
-1.25n + n2 = 125

The n term is -1.25n.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25n + 0.390625 + n2 = 125 + 0.390625

Reorder the terms:
0.390625 + -1.25n + n2 = 125 + 0.390625

Combine like terms: 125 + 0.390625 = 125.390625
0.390625 + -1.25n + n2 = 125.390625

Factor a perfect square on the left side:
(n + -0.625)(n + -0.625) = 125.390625

Calculate the square root of the right side: 11.197795542

Break this problem into two subproblems by setting 
(n + -0.625) equal to 11.197795542 and -11.197795542.

Subproblem 1

n + -0.625 = 11.197795542 Simplifying n + -0.625 = 11.197795542 Reorder the terms: -0.625 + n = 11.197795542 Solving -0.625 + n = 11.197795542 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = 11.197795542 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = 11.197795542 + 0.625 n = 11.197795542 + 0.625 Combine like terms: 11.197795542 + 0.625 = 11.822795542 n = 11.822795542 Simplifying n = 11.822795542

Subproblem 2

n + -0.625 = -11.197795542 Simplifying n + -0.625 = -11.197795542 Reorder the terms: -0.625 + n = -11.197795542 Solving -0.625 + n = -11.197795542 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = -11.197795542 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = -11.197795542 + 0.625 n = -11.197795542 + 0.625 Combine like terms: -11.197795542 + 0.625 = -10.572795542 n = -10.572795542 Simplifying n = -10.572795542

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.822795542, -10.572795542}

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