-4x^2-23x-6=0

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Solution for -4x^2-23x-6=0 equation: 


Simplifying
-4x2 + -23x + -6 = 0

Reorder the terms:
-6 + -23x + -4x2 = 0

Solving
-6 + -23x + -4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '-1'.
-1(6 + 23x + 4x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(6 + 23x + 4x2)' equal to zero and attempt to solve:

Simplifying
6 + 23x + 4x2 = 0

Solving
6 + 23x + 4x2 = 0

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

Divide each side by '4'.
1.5 + 5.75x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 5.75x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 5.75x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 5.75x + x2 = 0 + -1.5
5.75x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
5.75x + x2 = -1.5

The x term is 5.75x.  Take half its coefficient (2.875).
Square it (8.265625) and add it to both sides.

Add '8.265625' to each side of the equation.
5.75x + 8.265625 + x2 = -1.5 + 8.265625

Reorder the terms:
8.265625 + 5.75x + x2 = -1.5 + 8.265625

Combine like terms: -1.5 + 8.265625 = 6.765625
8.265625 + 5.75x + x2 = 6.765625

Factor a perfect square on the left side:
(x + 2.875)(x + 2.875) = 6.765625

Calculate the square root of the right side: 2.601081506

Break this problem into two subproblems by setting 
(x + 2.875) equal to 2.601081506 and -2.601081506.


Subproblem 1
x + 2.875 = 2.601081506

Simplifying
x + 2.875 = 2.601081506

Reorder the terms:
2.875 + x = 2.601081506

Solving
2.875 + x = 2.601081506

Solving for variable 'x'.

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

Add '-2.875' to each side of the equation.
2.875 + -2.875 + x = 2.601081506 + -2.875

Combine like terms: 2.875 + -2.875 = 0.000
0.000 + x = 2.601081506 + -2.875
x = 2.601081506 + -2.875

Combine like terms: 2.601081506 + -2.875 = -0.273918494
x = -0.273918494

Simplifying
x = -0.273918494


Subproblem 2
x + 2.875 = -2.601081506

Simplifying
x + 2.875 = -2.601081506

Reorder the terms:
2.875 + x = -2.601081506

Solving
2.875 + x = -2.601081506

Solving for variable 'x'.

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

Add '-2.875' to each side of the equation.
2.875 + -2.875 + x = -2.601081506 + -2.875

Combine like terms: 2.875 + -2.875 = 0.000
0.000 + x = -2.601081506 + -2.875
x = -2.601081506 + -2.875

Combine like terms: -2.601081506 + -2.875 = -5.476081506
x = -5.476081506

Simplifying
x = -5.476081506


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.273918494, -5.476081506}
Solution
x = {-0.273918494, -5.476081506}

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