-6x^2+42x+6=0

Solution for -6x^2+42x+6=0 equation:

Simplifying
-6x2 + 42x + 6 = 0

Reorder the terms:
6 + 42x + -6x2 = 0

Solving
6 + 42x + -6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(1 + 7x + -1x2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(1 + 7x + -1x2)' equal to zero and attempt to solve:

Simplifying
1 + 7x + -1x2 = 0

Solving
1 + 7x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -7x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -7x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -7x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -7x + x2 = 0 + 1
-7x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-7x + x2 = 1

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = 1 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 1 + 12.25

Combine like terms: 1 + 12.25 = 13.25
12.25 + -7x + x2 = 13.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(x + -3.5) equal to 3.640054945 and -3.640054945.

Subproblem 1
x + -3.5 = 3.640054945

Simplifying
x + -3.5 = 3.640054945

Reorder the terms:
-3.5 + x = 3.640054945

Solving
-3.5 + x = 3.640054945

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = 3.640054945 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 3.640054945 + 3.5
x = 3.640054945 + 3.5

Combine like terms: 3.640054945 + 3.5 = 7.140054945
x = 7.140054945

Simplifying
x = 7.140054945

Subproblem 2
x + -3.5 = -3.640054945

Simplifying
x + -3.5 = -3.640054945

Reorder the terms:
-3.5 + x = -3.640054945

Solving
-3.5 + x = -3.640054945

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = -3.640054945 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -3.640054945 + 3.5
x = -3.640054945 + 3.5

Combine like terms: -3.640054945 + 3.5 = -0.140054945
x = -0.140054945

Simplifying
x = -0.140054945

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.140054945, -0.140054945}
Solution
x = {7.140054945, -0.140054945}