6x^2+46x+24=0

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Solution for 6x^2+46x+24=0 equation: 


Simplifying
6x2 + 46x + 24 = 0

Reorder the terms:
24 + 46x + 6x2 = 0

Solving
24 + 46x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(12 + 23x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(12 + 23x + 3x2)' equal to zero and attempt to solve:

Simplifying
12 + 23x + 3x2 = 0

Solving
12 + 23x + 3x2 = 0

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

Divide each side by '3'.
4 + 7.666666667x + x2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 7.666666667x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + 7.666666667x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 7.666666667x + x2 = 0 + -4
7.666666667x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
7.666666667x + x2 = -4

The x term is 7.666666667x.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667x + 14.69444445 + x2 = -4 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667x + x2 = -4 + 14.69444445

Combine like terms: -4 + 14.69444445 = 10.69444445
14.69444445 + 7.666666667x + x2 = 10.69444445

Factor a perfect square on the left side:
(x + 3.833333334)(x + 3.833333334) = 10.69444445

Calculate the square root of the right side: 3.270236146

Break this problem into two subproblems by setting 
(x + 3.833333334) equal to 3.270236146 and -3.270236146.


Subproblem 1
x + 3.833333334 = 3.270236146

Simplifying
x + 3.833333334 = 3.270236146

Reorder the terms:
3.833333334 + x = 3.270236146

Solving
3.833333334 + x = 3.270236146

Solving for variable 'x'.

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

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + x = 3.270236146 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = 3.270236146 + -3.833333334
x = 3.270236146 + -3.833333334

Combine like terms: 3.270236146 + -3.833333334 = -0.563097188
x = -0.563097188

Simplifying
x = -0.563097188


Subproblem 2
x + 3.833333334 = -3.270236146

Simplifying
x + 3.833333334 = -3.270236146

Reorder the terms:
3.833333334 + x = -3.270236146

Solving
3.833333334 + x = -3.270236146

Solving for variable 'x'.

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

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + x = -3.270236146 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = -3.270236146 + -3.833333334
x = -3.270236146 + -3.833333334

Combine like terms: -3.270236146 + -3.833333334 = -7.10356948
x = -7.10356948

Simplifying
x = -7.10356948


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.563097188, -7.10356948}
Solution
x = {-0.563097188, -7.10356948}

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