(4x^2)-4x-38=0

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Solution for (4x^2)-4x-38=0 equation: 


Simplifying
(4x2) + -4x + -38 = 0

Reorder the terms:
-38 + -4x + (4x2) = 0

Solving
-38 + -4x + (4x2) = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-19 + -2x + (2x2)) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-19 + -2x + (2x2))' equal to zero and attempt to solve:

Simplifying
-19 + -2x + (2x2) = 0

Solving
-19 + -2x + (2x2) = 0

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

Divide each side by '2'.
-9.5 + -1x + x2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + -1x + 9.5 + x2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + -1x + x2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + -1x + x2 = 0 + 9.5
-1x + x2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
-1x + x2 = 9.5

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 9.5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 9.5 + 0.25

Combine like terms: 9.5 + 0.25 = 9.75
0.25 + -1x + x2 = 9.75

Factor a perfect square on the left side:
((x) + -0.5)((x) + -0.5) = 9.75

Calculate the square root of the right side: 3.122498999

Break this problem into two subproblems by setting 
((x) + -0.5) equal to 3.122498999 and -3.122498999.


Subproblem 1
(x) + -0.5 = 3.122498999

Simplifying
(x) + -0.5 = 3.122498999
x + -0.5 = 3.122498999

Reorder the terms:
-0.5 + x = 3.122498999

Solving
-0.5 + x = 3.122498999

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 3.122498999 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 3.122498999 + 0.5
x = 3.122498999 + 0.5

Combine like terms: 3.122498999 + 0.5 = 3.622498999
x = 3.622498999

Simplifying
x = 3.622498999


Subproblem 2
(x) + -0.5 = -3.122498999

Simplifying
(x) + -0.5 = -3.122498999
x + -0.5 = -3.122498999

Reorder the terms:
-0.5 + x = -3.122498999

Solving
-0.5 + x = -3.122498999

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -3.122498999 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -3.122498999 + 0.5
x = -3.122498999 + 0.5

Combine like terms: -3.122498999 + 0.5 = -2.622498999
x = -2.622498999

Simplifying
x = -2.622498999


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.622498999, -2.622498999}
Solution
x = {3.622498999, -2.622498999}

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