6x^2+52x+72=0

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Solution for 6x^2+52x+72=0 equation: 


Simplifying
6x2 + 52x + 72 = 0

Reorder the terms:
72 + 52x + 6x2 = 0

Solving
72 + 52x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(36 + 26x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(36 + 26x + 3x2)' equal to zero and attempt to solve:

Simplifying
36 + 26x + 3x2 = 0

Solving
36 + 26x + 3x2 = 0

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

Divide each side by '3'.
12 + 8.666666667x + x2 = 0

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + 8.666666667x + -12 + x2 = 0 + -12

Reorder the terms:
12 + -12 + 8.666666667x + x2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + 8.666666667x + x2 = 0 + -12
8.666666667x + x2 = 0 + -12

Combine like terms: 0 + -12 = -12
8.666666667x + x2 = -12

The x term is 8.666666667x.  Take half its coefficient (4.333333334).
Square it (18.77777778) and add it to both sides.

Add '18.77777778' to each side of the equation.
8.666666667x + 18.77777778 + x2 = -12 + 18.77777778

Reorder the terms:
18.77777778 + 8.666666667x + x2 = -12 + 18.77777778

Combine like terms: -12 + 18.77777778 = 6.77777778
18.77777778 + 8.666666667x + x2 = 6.77777778

Factor a perfect square on the left side:
(x + 4.333333334)(x + 4.333333334) = 6.77777778

Calculate the square root of the right side: 2.603416559

Break this problem into two subproblems by setting 
(x + 4.333333334) equal to 2.603416559 and -2.603416559.


Subproblem 1
x + 4.333333334 = 2.603416559

Simplifying
x + 4.333333334 = 2.603416559

Reorder the terms:
4.333333334 + x = 2.603416559

Solving
4.333333334 + x = 2.603416559

Solving for variable 'x'.

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

Add '-4.333333334' to each side of the equation.
4.333333334 + -4.333333334 + x = 2.603416559 + -4.333333334

Combine like terms: 4.333333334 + -4.333333334 = 0.000000000
0.000000000 + x = 2.603416559 + -4.333333334
x = 2.603416559 + -4.333333334

Combine like terms: 2.603416559 + -4.333333334 = -1.729916775
x = -1.729916775

Simplifying
x = -1.729916775


Subproblem 2
x + 4.333333334 = -2.603416559

Simplifying
x + 4.333333334 = -2.603416559

Reorder the terms:
4.333333334 + x = -2.603416559

Solving
4.333333334 + x = -2.603416559

Solving for variable 'x'.

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

Add '-4.333333334' to each side of the equation.
4.333333334 + -4.333333334 + x = -2.603416559 + -4.333333334

Combine like terms: 4.333333334 + -4.333333334 = 0.000000000
0.000000000 + x = -2.603416559 + -4.333333334
x = -2.603416559 + -4.333333334

Combine like terms: -2.603416559 + -4.333333334 = -6.936749893
x = -6.936749893

Simplifying
x = -6.936749893


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.729916775, -6.936749893}
Solution
x = {-1.729916775, -6.936749893}

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