6x^2+2x-52=0

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


Simplifying
6x2 + 2x + -52 = 0

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

Solving
-52 + 2x + 6x2 = 0

Solving for variable 'x'.

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

Ignore the factor 2.

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

Simplifying
-26 + x + 3x2 = 0

Solving
-26 + x + 3x2 = 0

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

Divide each side by '3'.
-8.666666667 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '8.666666667' to each side of the equation.
-8.666666667 + 0.3333333333x + 8.666666667 + x2 = 0 + 8.666666667

Reorder the terms:
-8.666666667 + 8.666666667 + 0.3333333333x + x2 = 0 + 8.666666667

Combine like terms: -8.666666667 + 8.666666667 = 0.000000000
0.000000000 + 0.3333333333x + x2 = 0 + 8.666666667
0.3333333333x + x2 = 0 + 8.666666667

Combine like terms: 0 + 8.666666667 = 8.666666667
0.3333333333x + x2 = 8.666666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333x + 0.25 + x2 = 8.666666667 + 0.25

Reorder the terms:
0.25 + 0.3333333333x + x2 = 8.666666667 + 0.25

Combine like terms: 8.666666667 + 0.25 = 8.916666667
0.25 + 0.3333333333x + x2 = 8.916666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 8.916666667

Calculate the square root of the right side: 2.986078811

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.986078811 and -2.986078811.


Subproblem 1
x + 0.5 = 2.986078811

Simplifying
x + 0.5 = 2.986078811

Reorder the terms:
0.5 + x = 2.986078811

Solving
0.5 + x = 2.986078811

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 2.986078811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.986078811 + -0.5
x = 2.986078811 + -0.5

Combine like terms: 2.986078811 + -0.5 = 2.486078811
x = 2.486078811

Simplifying
x = 2.486078811


Subproblem 2
x + 0.5 = -2.986078811

Simplifying
x + 0.5 = -2.986078811

Reorder the terms:
0.5 + x = -2.986078811

Solving
0.5 + x = -2.986078811

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -2.986078811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.986078811 + -0.5
x = -2.986078811 + -0.5

Combine like terms: -2.986078811 + -0.5 = -3.486078811
x = -3.486078811

Simplifying
x = -3.486078811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.486078811, -3.486078811}
Solution
x = {2.486078811, -3.486078811}

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