3x^2+23x+23=0

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Solution for 3x^2+23x+23=0 equation: 


Simplifying
3x2 + 23x + 23 = 0

Reorder the terms:
23 + 23x + 3x2 = 0

Solving
23 + 23x + 3x2 = 0

Solving for variable 'x'.

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

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

Move the constant term to the right:

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

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

Combine like terms: 7.666666667 + -7.666666667 = 0.000000000
0.000000000 + 7.666666667x + x2 = 0 + -7.666666667
7.666666667x + x2 = 0 + -7.666666667

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

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 = -7.666666667 + 14.69444445

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

Combine like terms: -7.666666667 + 14.69444445 = 7.027777783
14.69444445 + 7.666666667x + x2 = 7.027777783

Factor a perfect square on the left side:
(x + 3.833333334)(x + 3.833333334) = 7.027777783

Calculate the square root of the right side: 2.650995621

Break this problem into two subproblems by setting 
(x + 3.833333334) equal to 2.650995621 and -2.650995621.


Subproblem 1
x + 3.833333334 = 2.650995621

Simplifying
x + 3.833333334 = 2.650995621

Reorder the terms:
3.833333334 + x = 2.650995621

Solving
3.833333334 + x = 2.650995621

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 = 2.650995621 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = 2.650995621 + -3.833333334
x = 2.650995621 + -3.833333334

Combine like terms: 2.650995621 + -3.833333334 = -1.182337713
x = -1.182337713

Simplifying
x = -1.182337713


Subproblem 2
x + 3.833333334 = -2.650995621

Simplifying
x + 3.833333334 = -2.650995621

Reorder the terms:
3.833333334 + x = -2.650995621

Solving
3.833333334 + x = -2.650995621

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 = -2.650995621 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = -2.650995621 + -3.833333334
x = -2.650995621 + -3.833333334

Combine like terms: -2.650995621 + -3.833333334 = -6.484328955
x = -6.484328955

Simplifying
x = -6.484328955


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.182337713, -6.484328955}

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