6x^2+7x-78=0

Solution for 6x^2+7x-78=0 equation:

Simplifying
6x2 + 7x + -78 = 0

Reorder the terms:
-78 + 7x + 6x2 = 0

Solving
-78 + 7x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-13 + 1.166666667x + x2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 1.166666667x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + 1.166666667x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 1.166666667x + x2 = 0 + 13
1.166666667x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
1.166666667x + x2 = 13

The x term is 1.166666667x.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667x + 0.3402777780 + x2 = 13 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667x + x2 = 13 + 0.3402777780

Combine like terms: 13 + 0.3402777780 = 13.340277778
0.3402777780 + 1.166666667x + x2 = 13.340277778

Factor a perfect square on the left side:
(x + 0.5833333335)(x + 0.5833333335) = 13.340277778

Calculate the square root of the right side: 3.6524345

Break this problem into two subproblems by setting 
(x + 0.5833333335) equal to 3.6524345 and -3.6524345.

Subproblem 1
x + 0.5833333335 = 3.6524345

Simplifying
x + 0.5833333335 = 3.6524345

Reorder the terms:
0.5833333335 + x = 3.6524345

Solving
0.5833333335 + x = 3.6524345

Solving for variable 'x'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + x = 3.6524345 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + x = 3.6524345 + -0.5833333335
x = 3.6524345 + -0.5833333335

Combine like terms: 3.6524345 + -0.5833333335 = 3.0691011665
x = 3.0691011665

Simplifying
x = 3.0691011665

Subproblem 2
x + 0.5833333335 = -3.6524345

Simplifying
x + 0.5833333335 = -3.6524345

Reorder the terms:
0.5833333335 + x = -3.6524345

Solving
0.5833333335 + x = -3.6524345

Solving for variable 'x'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + x = -3.6524345 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + x = -3.6524345 + -0.5833333335
x = -3.6524345 + -0.5833333335

Combine like terms: -3.6524345 + -0.5833333335 = -4.2357678335
x = -4.2357678335

Simplifying
x = -4.2357678335

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.0691011665, -4.2357678335}