4x^2+52x+146=0

Solution for 4x^2+52x+146=0 equation:

Simplifying
4x2 + 52x + 146 = 0

Reorder the terms:
146 + 52x + 4x2 = 0

Solving
146 + 52x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(73 + 26x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(73 + 26x + 2x2)' equal to zero and attempt to solve:

Simplifying
73 + 26x + 2x2 = 0

Solving
73 + 26x + 2x2 = 0

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

Divide each side by '2'.
36.5 + 13x + x2 = 0

Move the constant term to the right:

Add '-36.5' to each side of the equation.
36.5 + 13x + -36.5 + x2 = 0 + -36.5

Reorder the terms:
36.5 + -36.5 + 13x + x2 = 0 + -36.5

Combine like terms: 36.5 + -36.5 = 0.0
0.0 + 13x + x2 = 0 + -36.5
13x + x2 = 0 + -36.5

Combine like terms: 0 + -36.5 = -36.5
13x + x2 = -36.5

The x term is 13x.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13x + 42.25 + x2 = -36.5 + 42.25

Reorder the terms:
42.25 + 13x + x2 = -36.5 + 42.25

Combine like terms: -36.5 + 42.25 = 5.75
42.25 + 13x + x2 = 5.75

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(x + 6.5) equal to 2.397915762 and -2.397915762.

Subproblem 1
x + 6.5 = 2.397915762

Simplifying
x + 6.5 = 2.397915762

Reorder the terms:
6.5 + x = 2.397915762

Solving
6.5 + x = 2.397915762

Solving for variable 'x'.

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

Add '-6.5' to each side of the equation.
6.5 + -6.5 + x = 2.397915762 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + x = 2.397915762 + -6.5
x = 2.397915762 + -6.5

Combine like terms: 2.397915762 + -6.5 = -4.102084238
x = -4.102084238

Simplifying
x = -4.102084238

Subproblem 2
x + 6.5 = -2.397915762

Simplifying
x + 6.5 = -2.397915762

Reorder the terms:
6.5 + x = -2.397915762

Solving
6.5 + x = -2.397915762

Solving for variable 'x'.

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

Add '-6.5' to each side of the equation.
6.5 + -6.5 + x = -2.397915762 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + x = -2.397915762 + -6.5
x = -2.397915762 + -6.5

Combine like terms: -2.397915762 + -6.5 = -8.897915762
x = -8.897915762

Simplifying
x = -8.897915762

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4.102084238, -8.897915762}
Solution
x = {-4.102084238, -8.897915762}