3x^2+23x-6.5=0

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Solution for 3x^2+23x-6.5=0 equation: 


Simplifying
3x2 + 23x + -6.5 = 0

Reorder the terms:
-6.5 + 23x + 3x2 = 0

Solving
-6.5 + 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'.
-2.166666667 + 7.666666667x + x2 = 0

Move the constant term to the right:

Add '2.166666667' to each side of the equation.
-2.166666667 + 7.666666667x + 2.166666667 + x2 = 0 + 2.166666667

Reorder the terms:
-2.166666667 + 2.166666667 + 7.666666667x + x2 = 0 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + 7.666666667x + x2 = 0 + 2.166666667
7.666666667x + x2 = 0 + 2.166666667

Combine like terms: 0 + 2.166666667 = 2.166666667
7.666666667x + x2 = 2.166666667

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 = 2.166666667 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667x + x2 = 2.166666667 + 14.69444445

Combine like terms: 2.166666667 + 14.69444445 = 16.861111117
14.69444445 + 7.666666667x + x2 = 16.861111117

Factor a perfect square on the left side:
(x + 3.833333334)(x + 3.833333334) = 16.861111117

Calculate the square root of the right side: 4.106228332

Break this problem into two subproblems by setting 
(x + 3.833333334) equal to 4.106228332 and -4.106228332.


Subproblem 1
x + 3.833333334 = 4.106228332

Simplifying
x + 3.833333334 = 4.106228332

Reorder the terms:
3.833333334 + x = 4.106228332

Solving
3.833333334 + x = 4.106228332

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 = 4.106228332 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = 4.106228332 + -3.833333334
x = 4.106228332 + -3.833333334

Combine like terms: 4.106228332 + -3.833333334 = 0.272894998
x = 0.272894998

Simplifying
x = 0.272894998


Subproblem 2
x + 3.833333334 = -4.106228332

Simplifying
x + 3.833333334 = -4.106228332

Reorder the terms:
3.833333334 + x = -4.106228332

Solving
3.833333334 + x = -4.106228332

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 = -4.106228332 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = -4.106228332 + -3.833333334
x = -4.106228332 + -3.833333334

Combine like terms: -4.106228332 + -3.833333334 = -7.939561666
x = -7.939561666

Simplifying
x = -7.939561666


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.272894998, -7.939561666}

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