3.1x^2-4.2x-7.8=0

Solution for 3.1x^2-4.2x-7.8=0 equation:

Simplifying
3.1x2 + -4.2x + -7.8 = 0

Reorder the terms:
-7.8 + -4.2x + 3.1x2 = 0

Solving
-7.8 + -4.2x + 3.1x2 = 0

Solving for variable 'x'.

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

Divide each side by '3.1'.
-2.516129032 + -1.35483871x + x2 = 0

Move the constant term to the right:

Add '2.516129032' to each side of the equation.
-2.516129032 + -1.35483871x + 2.516129032 + x2 = 0 + 2.516129032

Reorder the terms:
-2.516129032 + 2.516129032 + -1.35483871x + x2 = 0 + 2.516129032

Combine like terms: -2.516129032 + 2.516129032 = 0.000000000
0.000000000 + -1.35483871x + x2 = 0 + 2.516129032
-1.35483871x + x2 = 0 + 2.516129032

Combine like terms: 0 + 2.516129032 = 2.516129032
-1.35483871x + x2 = 2.516129032

The x term is -1.35483871x.  Take half its coefficient (-0.677419355).
Square it (0.4588969825) and add it to both sides.

Add '0.4588969825' to each side of the equation.
-1.35483871x + 0.4588969825 + x2 = 2.516129032 + 0.4588969825

Reorder the terms:
0.4588969825 + -1.35483871x + x2 = 2.516129032 + 0.4588969825

Combine like terms: 2.516129032 + 0.4588969825 = 2.9750260145
0.4588969825 + -1.35483871x + x2 = 2.9750260145

Factor a perfect square on the left side:
(x + -0.677419355)(x + -0.677419355) = 2.9750260145

Calculate the square root of the right side: 1.724826372

Break this problem into two subproblems by setting 
(x + -0.677419355) equal to 1.724826372 and -1.724826372.

Subproblem 1
x + -0.677419355 = 1.724826372

Simplifying
x + -0.677419355 = 1.724826372

Reorder the terms:
-0.677419355 + x = 1.724826372

Solving
-0.677419355 + x = 1.724826372

Solving for variable 'x'.

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

Add '0.677419355' to each side of the equation.
-0.677419355 + 0.677419355 + x = 1.724826372 + 0.677419355

Combine like terms: -0.677419355 + 0.677419355 = 0.000000000
0.000000000 + x = 1.724826372 + 0.677419355
x = 1.724826372 + 0.677419355

Combine like terms: 1.724826372 + 0.677419355 = 2.402245727
x = 2.402245727

Simplifying
x = 2.402245727

Subproblem 2
x + -0.677419355 = -1.724826372

Simplifying
x + -0.677419355 = -1.724826372

Reorder the terms:
-0.677419355 + x = -1.724826372

Solving
-0.677419355 + x = -1.724826372

Solving for variable 'x'.

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

Add '0.677419355' to each side of the equation.
-0.677419355 + 0.677419355 + x = -1.724826372 + 0.677419355

Combine like terms: -0.677419355 + 0.677419355 = 0.000000000
0.000000000 + x = -1.724826372 + 0.677419355
x = -1.724826372 + 0.677419355

Combine like terms: -1.724826372 + 0.677419355 = -1.047407017
x = -1.047407017

Simplifying
x = -1.047407017

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.402245727, -1.047407017}