3x^2+31x-6.7=0

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Solution for 3x^2+31x-6.7=0 equation: 


Simplifying
3x2 + 31x + -6.7 = 0

Reorder the terms:
-6.7 + 31x + 3x2 = 0

Solving
-6.7 + 31x + 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'.
-2.233333333 + 10.33333333x + x2 = 0

Move the constant term to the right:

Add '2.233333333' to each side of the equation.
-2.233333333 + 10.33333333x + 2.233333333 + x2 = 0 + 2.233333333

Reorder the terms:
-2.233333333 + 2.233333333 + 10.33333333x + x2 = 0 + 2.233333333

Combine like terms: -2.233333333 + 2.233333333 = 0.000000000
0.000000000 + 10.33333333x + x2 = 0 + 2.233333333
10.33333333x + x2 = 0 + 2.233333333

Combine like terms: 0 + 2.233333333 = 2.233333333
10.33333333x + x2 = 2.233333333

The x term is 10.33333333x.  Take half its coefficient (5.166666665).
Square it (26.69444443) and add it to both sides.

Add '26.69444443' to each side of the equation.
10.33333333x + 26.69444443 + x2 = 2.233333333 + 26.69444443

Reorder the terms:
26.69444443 + 10.33333333x + x2 = 2.233333333 + 26.69444443

Combine like terms: 2.233333333 + 26.69444443 = 28.927777763
26.69444443 + 10.33333333x + x2 = 28.927777763

Factor a perfect square on the left side:
(x + 5.166666665)(x + 5.166666665) = 28.927777763

Calculate the square root of the right side: 5.378454961

Break this problem into two subproblems by setting 
(x + 5.166666665) equal to 5.378454961 and -5.378454961.


Subproblem 1
x + 5.166666665 = 5.378454961

Simplifying
x + 5.166666665 = 5.378454961

Reorder the terms:
5.166666665 + x = 5.378454961

Solving
5.166666665 + x = 5.378454961

Solving for variable 'x'.

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

Add '-5.166666665' to each side of the equation.
5.166666665 + -5.166666665 + x = 5.378454961 + -5.166666665

Combine like terms: 5.166666665 + -5.166666665 = 0.000000000
0.000000000 + x = 5.378454961 + -5.166666665
x = 5.378454961 + -5.166666665

Combine like terms: 5.378454961 + -5.166666665 = 0.211788296
x = 0.211788296

Simplifying
x = 0.211788296


Subproblem 2
x + 5.166666665 = -5.378454961

Simplifying
x + 5.166666665 = -5.378454961

Reorder the terms:
5.166666665 + x = -5.378454961

Solving
5.166666665 + x = -5.378454961

Solving for variable 'x'.

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

Add '-5.166666665' to each side of the equation.
5.166666665 + -5.166666665 + x = -5.378454961 + -5.166666665

Combine like terms: 5.166666665 + -5.166666665 = 0.000000000
0.000000000 + x = -5.378454961 + -5.166666665
x = -5.378454961 + -5.166666665

Combine like terms: -5.378454961 + -5.166666665 = -10.545121626
x = -10.545121626

Simplifying
x = -10.545121626


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.211788296, -10.545121626}

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