6x^2-23x-7=0

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Solution for 6x^2-23x-7=0 equation: 


Simplifying
6x2 + -23x + -7 = 0

Reorder the terms:
-7 + -23x + 6x2 = 0

Solving
-7 + -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'.
-1.166666667 + -3.833333333x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -3.833333333x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -3.833333333x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -3.833333333x + x2 = 0 + 1.166666667
-3.833333333x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-3.833333333x + x2 = 1.166666667

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 = 1.166666667 + 3.673611112

Reorder the terms:
3.673611112 + -3.833333333x + x2 = 1.166666667 + 3.673611112

Combine like terms: 1.166666667 + 3.673611112 = 4.840277779
3.673611112 + -3.833333333x + x2 = 4.840277779

Factor a perfect square on the left side:
(x + -1.916666667)(x + -1.916666667) = 4.840277779

Calculate the square root of the right side: 2.200063131

Break this problem into two subproblems by setting 
(x + -1.916666667) equal to 2.200063131 and -2.200063131.


Subproblem 1
x + -1.916666667 = 2.200063131

Simplifying
x + -1.916666667 = 2.200063131

Reorder the terms:
-1.916666667 + x = 2.200063131

Solving
-1.916666667 + x = 2.200063131

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 = 2.200063131 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + x = 2.200063131 + 1.916666667
x = 2.200063131 + 1.916666667

Combine like terms: 2.200063131 + 1.916666667 = 4.116729798
x = 4.116729798

Simplifying
x = 4.116729798


Subproblem 2
x + -1.916666667 = -2.200063131

Simplifying
x + -1.916666667 = -2.200063131

Reorder the terms:
-1.916666667 + x = -2.200063131

Solving
-1.916666667 + x = -2.200063131

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 = -2.200063131 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + x = -2.200063131 + 1.916666667
x = -2.200063131 + 1.916666667

Combine like terms: -2.200063131 + 1.916666667 = -0.283396464
x = -0.283396464

Simplifying
x = -0.283396464


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.116729798, -0.283396464}

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