2n^2-2n+5=400

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Solution for 2n^2-2n+5=400 equation:


Simplifying
2n2 + -2n + 5 = 400

Reorder the terms:
5 + -2n + 2n2 = 400

Solving
5 + -2n + 2n2 = 400

Solving for variable 'n'.

Reorder the terms:
5 + -400 + -2n + 2n2 = 400 + -400

Combine like terms: 5 + -400 = -395
-395 + -2n + 2n2 = 400 + -400

Combine like terms: 400 + -400 = 0
-395 + -2n + 2n2 = 0

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

Divide each side by '2'.
-197.5 + -1n + n2 = 0

Move the constant term to the right:

Add '197.5' to each side of the equation.
-197.5 + -1n + 197.5 + n2 = 0 + 197.5

Reorder the terms:
-197.5 + 197.5 + -1n + n2 = 0 + 197.5

Combine like terms: -197.5 + 197.5 = 0.0
0.0 + -1n + n2 = 0 + 197.5
-1n + n2 = 0 + 197.5

Combine like terms: 0 + 197.5 = 197.5
-1n + n2 = 197.5

The n term is -1n.  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.
-1n + 0.25 + n2 = 197.5 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 197.5 + 0.25

Combine like terms: 197.5 + 0.25 = 197.75
0.25 + -1n + n2 = 197.75

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 197.75

Calculate the square root of the right side: 14.06236111

Break this problem into two subproblems by setting 
(n + -0.5) equal to 14.06236111 and -14.06236111.

Subproblem 1

n + -0.5 = 14.06236111 Simplifying n + -0.5 = 14.06236111 Reorder the terms: -0.5 + n = 14.06236111 Solving -0.5 + n = 14.06236111 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 = 14.06236111 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 14.06236111 + 0.5 n = 14.06236111 + 0.5 Combine like terms: 14.06236111 + 0.5 = 14.56236111 n = 14.56236111 Simplifying n = 14.56236111

Subproblem 2

n + -0.5 = -14.06236111 Simplifying n + -0.5 = -14.06236111 Reorder the terms: -0.5 + n = -14.06236111 Solving -0.5 + n = -14.06236111 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 = -14.06236111 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -14.06236111 + 0.5 n = -14.06236111 + 0.5 Combine like terms: -14.06236111 + 0.5 = -13.56236111 n = -13.56236111 Simplifying n = -13.56236111

Solution

The solution to the problem is based on the solutions from the subproblems. n = {14.56236111, -13.56236111}

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