2x^2-6x-52=0

Solution for 2x^2-6x-52=0 equation:

Simplifying
2x2 + -6x + -52 = 0

Reorder the terms:
-52 + -6x + 2x2 = 0

Solving
-52 + -6x + 2x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-26 + -3x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-26 + -3x + x2)' equal to zero and attempt to solve:

Simplifying
-26 + -3x + x2 = 0

Solving
-26 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '26' to each side of the equation.
-26 + -3x + 26 + x2 = 0 + 26

Reorder the terms:
-26 + 26 + -3x + x2 = 0 + 26

Combine like terms: -26 + 26 = 0
0 + -3x + x2 = 0 + 26
-3x + x2 = 0 + 26

Combine like terms: 0 + 26 = 26
-3x + x2 = 26

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 26 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 26 + 2.25

Combine like terms: 26 + 2.25 = 28.25
2.25 + -3x + x2 = 28.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 28.25

Calculate the square root of the right side: 5.315072906

Break this problem into two subproblems by setting 
(x + -1.5) equal to 5.315072906 and -5.315072906.

Subproblem 1
x + -1.5 = 5.315072906

Simplifying
x + -1.5 = 5.315072906

Reorder the terms:
-1.5 + x = 5.315072906

Solving
-1.5 + x = 5.315072906

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 5.315072906 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 5.315072906 + 1.5
x = 5.315072906 + 1.5

Combine like terms: 5.315072906 + 1.5 = 6.815072906
x = 6.815072906

Simplifying
x = 6.815072906

Subproblem 2
x + -1.5 = -5.315072906

Simplifying
x + -1.5 = -5.315072906

Reorder the terms:
-1.5 + x = -5.315072906

Solving
-1.5 + x = -5.315072906

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -5.315072906 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -5.315072906 + 1.5
x = -5.315072906 + 1.5

Combine like terms: -5.315072906 + 1.5 = -3.815072906
x = -3.815072906

Simplifying
x = -3.815072906

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.815072906, -3.815072906}
Solution
x = {6.815072906, -3.815072906}