2n^2+23n-420=0

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Solution for 2n^2+23n-420=0 equation:


Simplifying
2n2 + 23n + -420 = 0

Reorder the terms:
-420 + 23n + 2n2 = 0

Solving
-420 + 23n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
-210 + 11.5n + n2 = 0

Move the constant term to the right:

Add '210' to each side of the equation.
-210 + 11.5n + 210 + n2 = 0 + 210

Reorder the terms:
-210 + 210 + 11.5n + n2 = 0 + 210

Combine like terms: -210 + 210 = 0
0 + 11.5n + n2 = 0 + 210
11.5n + n2 = 0 + 210

Combine like terms: 0 + 210 = 210
11.5n + n2 = 210

The n term is 11.5n.  Take half its coefficient (5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
11.5n + 33.0625 + n2 = 210 + 33.0625

Reorder the terms:
33.0625 + 11.5n + n2 = 210 + 33.0625

Combine like terms: 210 + 33.0625 = 243.0625
33.0625 + 11.5n + n2 = 243.0625

Factor a perfect square on the left side:
(n + 5.75)(n + 5.75) = 243.0625

Calculate the square root of the right side: 15.590461828

Break this problem into two subproblems by setting 
(n + 5.75) equal to 15.590461828 and -15.590461828.

Subproblem 1

n + 5.75 = 15.590461828 Simplifying n + 5.75 = 15.590461828 Reorder the terms: 5.75 + n = 15.590461828 Solving 5.75 + n = 15.590461828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + n = 15.590461828 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + n = 15.590461828 + -5.75 n = 15.590461828 + -5.75 Combine like terms: 15.590461828 + -5.75 = 9.840461828 n = 9.840461828 Simplifying n = 9.840461828

Subproblem 2

n + 5.75 = -15.590461828 Simplifying n + 5.75 = -15.590461828 Reorder the terms: 5.75 + n = -15.590461828 Solving 5.75 + n = -15.590461828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + n = -15.590461828 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + n = -15.590461828 + -5.75 n = -15.590461828 + -5.75 Combine like terms: -15.590461828 + -5.75 = -21.340461828 n = -21.340461828 Simplifying n = -21.340461828

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.840461828, -21.340461828}

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