3x^2-43x-23=0

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Solution for 3x^2-43x-23=0 equation: 


Simplifying
3x2 + -43x + -23 = 0

Reorder the terms:
-23 + -43x + 3x2 = 0

Solving
-23 + -43x + 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 + -14.33333333x + x2 = 0

Move the constant term to the right:

Add '7.666666667' to each side of the equation.
-7.666666667 + -14.33333333x + 7.666666667 + x2 = 0 + 7.666666667

Reorder the terms:
-7.666666667 + 7.666666667 + -14.33333333x + x2 = 0 + 7.666666667

Combine like terms: -7.666666667 + 7.666666667 = 0.000000000
0.000000000 + -14.33333333x + x2 = 0 + 7.666666667
-14.33333333x + x2 = 0 + 7.666666667

Combine like terms: 0 + 7.666666667 = 7.666666667
-14.33333333x + x2 = 7.666666667

The x term is -14.33333333x.  Take half its coefficient (-7.166666665).
Square it (51.36111109) and add it to both sides.

Add '51.36111109' to each side of the equation.
-14.33333333x + 51.36111109 + x2 = 7.666666667 + 51.36111109

Reorder the terms:
51.36111109 + -14.33333333x + x2 = 7.666666667 + 51.36111109

Combine like terms: 7.666666667 + 51.36111109 = 59.027777757
51.36111109 + -14.33333333x + x2 = 59.027777757

Factor a perfect square on the left side:
(x + -7.166666665)(x + -7.166666665) = 59.027777757

Calculate the square root of the right side: 7.682953713

Break this problem into two subproblems by setting 
(x + -7.166666665) equal to 7.682953713 and -7.682953713.


Subproblem 1
x + -7.166666665 = 7.682953713

Simplifying
x + -7.166666665 = 7.682953713

Reorder the terms:
-7.166666665 + x = 7.682953713

Solving
-7.166666665 + x = 7.682953713

Solving for variable 'x'.

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

Add '7.166666665' to each side of the equation.
-7.166666665 + 7.166666665 + x = 7.682953713 + 7.166666665

Combine like terms: -7.166666665 + 7.166666665 = 0.000000000
0.000000000 + x = 7.682953713 + 7.166666665
x = 7.682953713 + 7.166666665

Combine like terms: 7.682953713 + 7.166666665 = 14.849620378
x = 14.849620378

Simplifying
x = 14.849620378


Subproblem 2
x + -7.166666665 = -7.682953713

Simplifying
x + -7.166666665 = -7.682953713

Reorder the terms:
-7.166666665 + x = -7.682953713

Solving
-7.166666665 + x = -7.682953713

Solving for variable 'x'.

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

Add '7.166666665' to each side of the equation.
-7.166666665 + 7.166666665 + x = -7.682953713 + 7.166666665

Combine like terms: -7.166666665 + 7.166666665 = 0.000000000
0.000000000 + x = -7.682953713 + 7.166666665
x = -7.682953713 + 7.166666665

Combine like terms: -7.682953713 + 7.166666665 = -0.516287048
x = -0.516287048

Simplifying
x = -0.516287048


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {14.849620378, -0.516287048}

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