31x^2+75.88x+42.66=0

Solution for 31x^2+75.88x+42.66=0 equation:

Simplifying
31x2 + 75.88x + 42.66 = 0

Reorder the terms:
42.66 + 75.88x + 31x2 = 0

Solving
42.66 + 75.88x + 31x2 = 0

Solving for variable 'x'.

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

Divide each side by '31'.
1.376129032 + 2.447741935x + x2 = 0

Move the constant term to the right:

Add '-1.376129032' to each side of the equation.
1.376129032 + 2.447741935x + -1.376129032 + x2 = 0 + -1.376129032

Reorder the terms:
1.376129032 + -1.376129032 + 2.447741935x + x2 = 0 + -1.376129032

Combine like terms: 1.376129032 + -1.376129032 = 0.000000000
0.000000000 + 2.447741935x + x2 = 0 + -1.376129032
2.447741935x + x2 = 0 + -1.376129032

Combine like terms: 0 + -1.376129032 = -1.376129032
2.447741935x + x2 = -1.376129032

The x term is 2.447741935x.  Take half its coefficient (1.223870968).
Square it (1.497860146) and add it to both sides.

Add '1.497860146' to each side of the equation.
2.447741935x + 1.497860146 + x2 = -1.376129032 + 1.497860146

Reorder the terms:
1.497860146 + 2.447741935x + x2 = -1.376129032 + 1.497860146

Combine like terms: -1.376129032 + 1.497860146 = 0.121731114
1.497860146 + 2.447741935x + x2 = 0.121731114

Factor a perfect square on the left side:
(x + 1.223870968)(x + 1.223870968) = 0.121731114

Calculate the square root of the right side: 0.348899862

Break this problem into two subproblems by setting 
(x + 1.223870968) equal to 0.348899862 and -0.348899862.

Subproblem 1
x + 1.223870968 = 0.348899862

Simplifying
x + 1.223870968 = 0.348899862

Reorder the terms:
1.223870968 + x = 0.348899862

Solving
1.223870968 + x = 0.348899862

Solving for variable 'x'.

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

Add '-1.223870968' to each side of the equation.
1.223870968 + -1.223870968 + x = 0.348899862 + -1.223870968

Combine like terms: 1.223870968 + -1.223870968 = 0.000000000
0.000000000 + x = 0.348899862 + -1.223870968
x = 0.348899862 + -1.223870968

Combine like terms: 0.348899862 + -1.223870968 = -0.874971106
x = -0.874971106

Simplifying
x = -0.874971106

Subproblem 2
x + 1.223870968 = -0.348899862

Simplifying
x + 1.223870968 = -0.348899862

Reorder the terms:
1.223870968 + x = -0.348899862

Solving
1.223870968 + x = -0.348899862

Solving for variable 'x'.

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

Add '-1.223870968' to each side of the equation.
1.223870968 + -1.223870968 + x = -0.348899862 + -1.223870968

Combine like terms: 1.223870968 + -1.223870968 = 0.000000000
0.000000000 + x = -0.348899862 + -1.223870968
x = -0.348899862 + -1.223870968

Combine like terms: -0.348899862 + -1.223870968 = -1.57277083
x = -1.57277083

Simplifying
x = -1.57277083

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.874971106, -1.57277083}