3n^2+77n-6500=0

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Solution for 3n^2+77n-6500=0 equation:


Simplifying
3n2 + 77n + -6500 = 0

Reorder the terms:
-6500 + 77n + 3n2 = 0

Solving
-6500 + 77n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-2166.666667 + 25.66666667n + n2 = 0

Move the constant term to the right:

Add '2166.666667' to each side of the equation.
-2166.666667 + 25.66666667n + 2166.666667 + n2 = 0 + 2166.666667

Reorder the terms:
-2166.666667 + 2166.666667 + 25.66666667n + n2 = 0 + 2166.666667

Combine like terms: -2166.666667 + 2166.666667 = 0.000000
0.000000 + 25.66666667n + n2 = 0 + 2166.666667
25.66666667n + n2 = 0 + 2166.666667

Combine like terms: 0 + 2166.666667 = 2166.666667
25.66666667n + n2 = 2166.666667

The n term is 25.66666667n.  Take half its coefficient (12.83333334).
Square it (164.6944446) and add it to both sides.

Add '164.6944446' to each side of the equation.
25.66666667n + 164.6944446 + n2 = 2166.666667 + 164.6944446

Reorder the terms:
164.6944446 + 25.66666667n + n2 = 2166.666667 + 164.6944446

Combine like terms: 2166.666667 + 164.6944446 = 2331.3611116
164.6944446 + 25.66666667n + n2 = 2331.3611116

Factor a perfect square on the left side:
(n + 12.83333334)(n + 12.83333334) = 2331.3611116

Calculate the square root of the right side: 48.284170404

Break this problem into two subproblems by setting 
(n + 12.83333334) equal to 48.284170404 and -48.284170404.

Subproblem 1

n + 12.83333334 = 48.284170404 Simplifying n + 12.83333334 = 48.284170404 Reorder the terms: 12.83333334 + n = 48.284170404 Solving 12.83333334 + n = 48.284170404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.83333334' to each side of the equation. 12.83333334 + -12.83333334 + n = 48.284170404 + -12.83333334 Combine like terms: 12.83333334 + -12.83333334 = 0.00000000 0.00000000 + n = 48.284170404 + -12.83333334 n = 48.284170404 + -12.83333334 Combine like terms: 48.284170404 + -12.83333334 = 35.450837064 n = 35.450837064 Simplifying n = 35.450837064

Subproblem 2

n + 12.83333334 = -48.284170404 Simplifying n + 12.83333334 = -48.284170404 Reorder the terms: 12.83333334 + n = -48.284170404 Solving 12.83333334 + n = -48.284170404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.83333334' to each side of the equation. 12.83333334 + -12.83333334 + n = -48.284170404 + -12.83333334 Combine like terms: 12.83333334 + -12.83333334 = 0.00000000 0.00000000 + n = -48.284170404 + -12.83333334 n = -48.284170404 + -12.83333334 Combine like terms: -48.284170404 + -12.83333334 = -61.117503744 n = -61.117503744 Simplifying n = -61.117503744

Solution

The solution to the problem is based on the solutions from the subproblems. n = {35.450837064, -61.117503744}

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