4x^2+52x-13=0

Solution for 4x^2+52x-13=0 equation:

Simplifying
4x2 + 52x + -13 = 0

Reorder the terms:
-13 + 52x + 4x2 = 0

Solving
-13 + 52x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-3.25 + 13x + x2 = 0

Move the constant term to the right:

Add '3.25' to each side of the equation.
-3.25 + 13x + 3.25 + x2 = 0 + 3.25

Reorder the terms:
-3.25 + 3.25 + 13x + x2 = 0 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + 13x + x2 = 0 + 3.25
13x + x2 = 0 + 3.25

Combine like terms: 0 + 3.25 = 3.25
13x + x2 = 3.25

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 = 3.25 + 42.25

Reorder the terms:
42.25 + 13x + x2 = 3.25 + 42.25

Combine like terms: 3.25 + 42.25 = 45.5
42.25 + 13x + x2 = 45.5

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 45.5

Calculate the square root of the right side: 6.745368782

Break this problem into two subproblems by setting 
(x + 6.5) equal to 6.745368782 and -6.745368782.

Subproblem 1
x + 6.5 = 6.745368782

Simplifying
x + 6.5 = 6.745368782

Reorder the terms:
6.5 + x = 6.745368782

Solving
6.5 + x = 6.745368782

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 = 6.745368782 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + x = 6.745368782 + -6.5
x = 6.745368782 + -6.5

Combine like terms: 6.745368782 + -6.5 = 0.245368782
x = 0.245368782

Simplifying
x = 0.245368782

Subproblem 2
x + 6.5 = -6.745368782

Simplifying
x + 6.5 = -6.745368782

Reorder the terms:
6.5 + x = -6.745368782

Solving
6.5 + x = -6.745368782

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 = -6.745368782 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + x = -6.745368782 + -6.5
x = -6.745368782 + -6.5

Combine like terms: -6.745368782 + -6.5 = -13.245368782
x = -13.245368782

Simplifying
x = -13.245368782

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.245368782, -13.245368782}