6x^2+58x+105=0

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Solution for 6x^2+58x+105=0 equation: 


Simplifying
6x2 + 58x + 105 = 0

Reorder the terms:
105 + 58x + 6x2 = 0

Solving
105 + 58x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
17.5 + 9.666666667x + x2 = 0

Move the constant term to the right:

Add '-17.5' to each side of the equation.
17.5 + 9.666666667x + -17.5 + x2 = 0 + -17.5

Reorder the terms:
17.5 + -17.5 + 9.666666667x + x2 = 0 + -17.5

Combine like terms: 17.5 + -17.5 = 0.0
0.0 + 9.666666667x + x2 = 0 + -17.5
9.666666667x + x2 = 0 + -17.5

Combine like terms: 0 + -17.5 = -17.5
9.666666667x + x2 = -17.5

The x term is 9.666666667x.  Take half its coefficient (4.833333334).
Square it (23.36111112) and add it to both sides.

Add '23.36111112' to each side of the equation.
9.666666667x + 23.36111112 + x2 = -17.5 + 23.36111112

Reorder the terms:
23.36111112 + 9.666666667x + x2 = -17.5 + 23.36111112

Combine like terms: -17.5 + 23.36111112 = 5.86111112
23.36111112 + 9.666666667x + x2 = 5.86111112

Factor a perfect square on the left side:
(x + 4.833333334)(x + 4.833333334) = 5.86111112

Calculate the square root of the right side: 2.420973176

Break this problem into two subproblems by setting 
(x + 4.833333334) equal to 2.420973176 and -2.420973176.


Subproblem 1
x + 4.833333334 = 2.420973176

Simplifying
x + 4.833333334 = 2.420973176

Reorder the terms:
4.833333334 + x = 2.420973176

Solving
4.833333334 + x = 2.420973176

Solving for variable 'x'.

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

Add '-4.833333334' to each side of the equation.
4.833333334 + -4.833333334 + x = 2.420973176 + -4.833333334

Combine like terms: 4.833333334 + -4.833333334 = 0.000000000
0.000000000 + x = 2.420973176 + -4.833333334
x = 2.420973176 + -4.833333334

Combine like terms: 2.420973176 + -4.833333334 = -2.412360158
x = -2.412360158

Simplifying
x = -2.412360158


Subproblem 2
x + 4.833333334 = -2.420973176

Simplifying
x + 4.833333334 = -2.420973176

Reorder the terms:
4.833333334 + x = -2.420973176

Solving
4.833333334 + x = -2.420973176

Solving for variable 'x'.

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

Add '-4.833333334' to each side of the equation.
4.833333334 + -4.833333334 + x = -2.420973176 + -4.833333334

Combine like terms: 4.833333334 + -4.833333334 = 0.000000000
0.000000000 + x = -2.420973176 + -4.833333334
x = -2.420973176 + -4.833333334

Combine like terms: -2.420973176 + -4.833333334 = -7.25430651
x = -7.25430651

Simplifying
x = -7.25430651


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.412360158, -7.25430651}

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