6x^2+23x+4=0

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Solution for 6x^2+23x+4=0 equation: 


Simplifying
6x2 + 23x + 4 = 0

Reorder the terms:
4 + 23x + 6x2 = 0

Solving
4 + 23x + 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'.
0.6666666667 + 3.833333333x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 3.833333333x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 3.833333333x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 3.833333333x + x2 = 0 + -0.6666666667
3.833333333x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
3.833333333x + x2 = -0.6666666667

The x term is 3.833333333x.  Take half its coefficient (1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
3.833333333x + 3.673611112 + x2 = -0.6666666667 + 3.673611112

Reorder the terms:
3.673611112 + 3.833333333x + x2 = -0.6666666667 + 3.673611112

Combine like terms: -0.6666666667 + 3.673611112 = 3.0069444453
3.673611112 + 3.833333333x + x2 = 3.0069444453

Factor a perfect square on the left side:
(x + 1.916666667)(x + 1.916666667) = 3.0069444453

Calculate the square root of the right side: 1.734054337

Break this problem into two subproblems by setting 
(x + 1.916666667) equal to 1.734054337 and -1.734054337.


Subproblem 1
x + 1.916666667 = 1.734054337

Simplifying
x + 1.916666667 = 1.734054337

Reorder the terms:
1.916666667 + x = 1.734054337

Solving
1.916666667 + x = 1.734054337

Solving for variable 'x'.

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

Add '-1.916666667' to each side of the equation.
1.916666667 + -1.916666667 + x = 1.734054337 + -1.916666667

Combine like terms: 1.916666667 + -1.916666667 = 0.000000000
0.000000000 + x = 1.734054337 + -1.916666667
x = 1.734054337 + -1.916666667

Combine like terms: 1.734054337 + -1.916666667 = -0.18261233
x = -0.18261233

Simplifying
x = -0.18261233


Subproblem 2
x + 1.916666667 = -1.734054337

Simplifying
x + 1.916666667 = -1.734054337

Reorder the terms:
1.916666667 + x = -1.734054337

Solving
1.916666667 + x = -1.734054337

Solving for variable 'x'.

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

Add '-1.916666667' to each side of the equation.
1.916666667 + -1.916666667 + x = -1.734054337 + -1.916666667

Combine like terms: 1.916666667 + -1.916666667 = 0.000000000
0.000000000 + x = -1.734054337 + -1.916666667
x = -1.734054337 + -1.916666667

Combine like terms: -1.734054337 + -1.916666667 = -3.650721004
x = -3.650721004

Simplifying
x = -3.650721004


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.18261233, -3.650721004}

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