4x^2-6x-52=0

Solution for 4x^2-6x-52=0 equation:

Simplifying
4x2 + -6x + -52 = 0

Reorder the terms:
-52 + -6x + 4x2 = 0

Solving
-52 + -6x + 4x2 = 0

Solving for variable 'x'.

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

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

Simplifying
-26 + -3x + 2x2 = 0

Solving
-26 + -3x + 2x2 = 0

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

Divide each side by '2'.
-13 + -1.5x + x2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -1.5x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + -1.5x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -1.5x + x2 = 0 + 13
-1.5x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
-1.5x + x2 = 13

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 13 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 13 + 0.5625

Combine like terms: 13 + 0.5625 = 13.5625
0.5625 + -1.5x + x2 = 13.5625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 13.5625

Calculate the square root of the right side: 3.682729966

Break this problem into two subproblems by setting 
(x + -0.75) equal to 3.682729966 and -3.682729966.

Subproblem 1
x + -0.75 = 3.682729966

Simplifying
x + -0.75 = 3.682729966

Reorder the terms:
-0.75 + x = 3.682729966

Solving
-0.75 + x = 3.682729966

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 3.682729966 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 3.682729966 + 0.75
x = 3.682729966 + 0.75

Combine like terms: 3.682729966 + 0.75 = 4.432729966
x = 4.432729966

Simplifying
x = 4.432729966

Subproblem 2
x + -0.75 = -3.682729966

Simplifying
x + -0.75 = -3.682729966

Reorder the terms:
-0.75 + x = -3.682729966

Solving
-0.75 + x = -3.682729966

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -3.682729966 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -3.682729966 + 0.75
x = -3.682729966 + 0.75

Combine like terms: -3.682729966 + 0.75 = -2.932729966
x = -2.932729966

Simplifying
x = -2.932729966

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.432729966, -2.932729966}
Solution
x = {4.432729966, -2.932729966}