3x^2+41x-180=0

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Solution for 3x^2+41x-180=0 equation: 


Simplifying
3x2 + 41x + -180 = 0

Reorder the terms:
-180 + 41x + 3x2 = 0

Solving
-180 + 41x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-60 + 13.66666667x + x2 = 0

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + 13.66666667x + 60 + x2 = 0 + 60

Reorder the terms:
-60 + 60 + 13.66666667x + x2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + 13.66666667x + x2 = 0 + 60
13.66666667x + x2 = 0 + 60

Combine like terms: 0 + 60 = 60
13.66666667x + x2 = 60

The x term is 13.66666667x.  Take half its coefficient (6.833333335).
Square it (46.69444447) and add it to both sides.

Add '46.69444447' to each side of the equation.
13.66666667x + 46.69444447 + x2 = 60 + 46.69444447

Reorder the terms:
46.69444447 + 13.66666667x + x2 = 60 + 46.69444447

Combine like terms: 60 + 46.69444447 = 106.69444447
46.69444447 + 13.66666667x + x2 = 106.69444447

Factor a perfect square on the left side:
(x + 6.833333335)(x + 6.833333335) = 106.69444447

Calculate the square root of the right side: 10.329300289

Break this problem into two subproblems by setting 
(x + 6.833333335) equal to 10.329300289 and -10.329300289.


Subproblem 1
x + 6.833333335 = 10.329300289

Simplifying
x + 6.833333335 = 10.329300289

Reorder the terms:
6.833333335 + x = 10.329300289

Solving
6.833333335 + x = 10.329300289

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-6.833333335' to each side of the equation.
6.833333335 + -6.833333335 + x = 10.329300289 + -6.833333335

Combine like terms: 6.833333335 + -6.833333335 = 0.000000000
0.000000000 + x = 10.329300289 + -6.833333335
x = 10.329300289 + -6.833333335

Combine like terms: 10.329300289 + -6.833333335 = 3.495966954
x = 3.495966954

Simplifying
x = 3.495966954


Subproblem 2
x + 6.833333335 = -10.329300289

Simplifying
x + 6.833333335 = -10.329300289

Reorder the terms:
6.833333335 + x = -10.329300289

Solving
6.833333335 + x = -10.329300289

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-6.833333335' to each side of the equation.
6.833333335 + -6.833333335 + x = -10.329300289 + -6.833333335

Combine like terms: 6.833333335 + -6.833333335 = 0.000000000
0.000000000 + x = -10.329300289 + -6.833333335
x = -10.329300289 + -6.833333335

Combine like terms: -10.329300289 + -6.833333335 = -17.162633624
x = -17.162633624

Simplifying
x = -17.162633624


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.495966954, -17.162633624}

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