0=n^2-40n-200

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Solution for 0=n^2-40n-200 equation:


Simplifying
0 = n2 + -40n + -200

Reorder the terms:
0 = -200 + -40n + n2

Solving
0 = -200 + -40n + n2

Solving for variable 'n'.

Combine like terms: 0 + 200 = 200
200 + 40n + -1n2 = -200 + -40n + n2 + 200 + 40n + -1n2

Reorder the terms:
200 + 40n + -1n2 = -200 + 200 + -40n + 40n + n2 + -1n2

Combine like terms: -200 + 200 = 0
200 + 40n + -1n2 = 0 + -40n + 40n + n2 + -1n2
200 + 40n + -1n2 = -40n + 40n + n2 + -1n2

Combine like terms: -40n + 40n = 0
200 + 40n + -1n2 = 0 + n2 + -1n2
200 + 40n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
200 + 40n + -1n2 = 0

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

Divide each side by '-1'.
-200 + -40n + n2 = 0

Move the constant term to the right:

Add '200' to each side of the equation.
-200 + -40n + 200 + n2 = 0 + 200

Reorder the terms:
-200 + 200 + -40n + n2 = 0 + 200

Combine like terms: -200 + 200 = 0
0 + -40n + n2 = 0 + 200
-40n + n2 = 0 + 200

Combine like terms: 0 + 200 = 200
-40n + n2 = 200

The n term is -40n.  Take half its coefficient (-20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
-40n + 400 + n2 = 200 + 400

Reorder the terms:
400 + -40n + n2 = 200 + 400

Combine like terms: 200 + 400 = 600
400 + -40n + n2 = 600

Factor a perfect square on the left side:
(n + -20)(n + -20) = 600

Calculate the square root of the right side: 24.494897428

Break this problem into two subproblems by setting 
(n + -20) equal to 24.494897428 and -24.494897428.

Subproblem 1

n + -20 = 24.494897428 Simplifying n + -20 = 24.494897428 Reorder the terms: -20 + n = 24.494897428 Solving -20 + n = 24.494897428 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + n = 24.494897428 + 20 Combine like terms: -20 + 20 = 0 0 + n = 24.494897428 + 20 n = 24.494897428 + 20 Combine like terms: 24.494897428 + 20 = 44.494897428 n = 44.494897428 Simplifying n = 44.494897428

Subproblem 2

n + -20 = -24.494897428 Simplifying n + -20 = -24.494897428 Reorder the terms: -20 + n = -24.494897428 Solving -20 + n = -24.494897428 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + n = -24.494897428 + 20 Combine like terms: -20 + 20 = 0 0 + n = -24.494897428 + 20 n = -24.494897428 + 20 Combine like terms: -24.494897428 + 20 = -4.494897428 n = -4.494897428 Simplifying n = -4.494897428

Solution

The solution to the problem is based on the solutions from the subproblems. n = {44.494897428, -4.494897428}

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