2x^2+3x-52=0

Solution for 2x^2+3x-52=0 equation:

Simplifying
2x2 + 3x + -52 = 0

Reorder the terms:
-52 + 3x + 2x2 = 0

Solving
-52 + 3x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-26 + 1.5x + x2 = 0

Move the constant term to the right:

Add '26' to each side of the equation.
-26 + 1.5x + 26 + x2 = 0 + 26

Reorder the terms:
-26 + 26 + 1.5x + x2 = 0 + 26

Combine like terms: -26 + 26 = 0
0 + 1.5x + x2 = 0 + 26
1.5x + x2 = 0 + 26

Combine like terms: 0 + 26 = 26
1.5x + x2 = 26

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 = 26 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 26 + 0.5625

Combine like terms: 26 + 0.5625 = 26.5625
0.5625 + 1.5x + x2 = 26.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 26.5625

Calculate the square root of the right side: 5.153882032

Break this problem into two subproblems by setting 
(x + 0.75) equal to 5.153882032 and -5.153882032.

Subproblem 1
x + 0.75 = 5.153882032

Simplifying
x + 0.75 = 5.153882032

Reorder the terms:
0.75 + x = 5.153882032

Solving
0.75 + x = 5.153882032

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 = 5.153882032 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 5.153882032 + -0.75
x = 5.153882032 + -0.75

Combine like terms: 5.153882032 + -0.75 = 4.403882032
x = 4.403882032

Simplifying
x = 4.403882032

Subproblem 2
x + 0.75 = -5.153882032

Simplifying
x + 0.75 = -5.153882032

Reorder the terms:
0.75 + x = -5.153882032

Solving
0.75 + x = -5.153882032

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 = -5.153882032 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -5.153882032 + -0.75
x = -5.153882032 + -0.75

Combine like terms: -5.153882032 + -0.75 = -5.903882032
x = -5.903882032

Simplifying
x = -5.903882032

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.403882032, -5.903882032}