3x^2-24x+x+2=0

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Solution for 3x^2-24x+x+2=0 equation: 


Simplifying
3x2 + -24x + x + 2 = 0

Reorder the terms:
2 + -24x + x + 3x2 = 0

Combine like terms: -24x + x = -23x
2 + -23x + 3x2 = 0

Solving
2 + -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'.
0.6666666667 + -7.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + -7.666666667x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + -7.666666667x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + -7.666666667x + x2 = 0 + -0.6666666667
-7.666666667x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
-7.666666667x + x2 = -0.6666666667

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 = -0.6666666667 + 14.69444445

Reorder the terms:
14.69444445 + -7.666666667x + x2 = -0.6666666667 + 14.69444445

Combine like terms: -0.6666666667 + 14.69444445 = 14.0277777833
14.69444445 + -7.666666667x + x2 = 14.0277777833

Factor a perfect square on the left side:
(x + -3.833333334)(x + -3.833333334) = 14.0277777833

Calculate the square root of the right side: 3.74536751

Break this problem into two subproblems by setting 
(x + -3.833333334) equal to 3.74536751 and -3.74536751.


Subproblem 1
x + -3.833333334 = 3.74536751

Simplifying
x + -3.833333334 = 3.74536751

Reorder the terms:
-3.833333334 + x = 3.74536751

Solving
-3.833333334 + x = 3.74536751

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 = 3.74536751 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + x = 3.74536751 + 3.833333334
x = 3.74536751 + 3.833333334

Combine like terms: 3.74536751 + 3.833333334 = 7.578700844
x = 7.578700844

Simplifying
x = 7.578700844


Subproblem 2
x + -3.833333334 = -3.74536751

Simplifying
x + -3.833333334 = -3.74536751

Reorder the terms:
-3.833333334 + x = -3.74536751

Solving
-3.833333334 + x = -3.74536751

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 = -3.74536751 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + x = -3.74536751 + 3.833333334
x = -3.74536751 + 3.833333334

Combine like terms: -3.74536751 + 3.833333334 = 0.087965824
x = 0.087965824

Simplifying
x = 0.087965824


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.578700844, 0.087965824}

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