4x^2-2x-32=0

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Solution for 4x^2-2x-32=0 equation: 


Simplifying
4x2 + -2x + -32 = 0

Reorder the terms:
-32 + -2x + 4x2 = 0

Solving
-32 + -2x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-16 + -1x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-16 + -1x + 2x2)' equal to zero and attempt to solve:

Simplifying
-16 + -1x + 2x2 = 0

Solving
-16 + -1x + 2x2 = 0

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

Divide each side by '2'.
-8 + -0.5x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -0.5x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + -0.5x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -0.5x + x2 = 0 + 8
-0.5x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
-0.5x + x2 = 8

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 8 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 8 + 0.0625

Combine like terms: 8 + 0.0625 = 8.0625
0.0625 + -0.5x + x2 = 8.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 8.0625

Calculate the square root of the right side: 2.839454173

Break this problem into two subproblems by setting 
(x + -0.25) equal to 2.839454173 and -2.839454173.


Subproblem 1
x + -0.25 = 2.839454173

Simplifying
x + -0.25 = 2.839454173

Reorder the terms:
-0.25 + x = 2.839454173

Solving
-0.25 + x = 2.839454173

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 2.839454173 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 2.839454173 + 0.25
x = 2.839454173 + 0.25

Combine like terms: 2.839454173 + 0.25 = 3.089454173
x = 3.089454173

Simplifying
x = 3.089454173


Subproblem 2
x + -0.25 = -2.839454173

Simplifying
x + -0.25 = -2.839454173

Reorder the terms:
-0.25 + x = -2.839454173

Solving
-0.25 + x = -2.839454173

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -2.839454173 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -2.839454173 + 0.25
x = -2.839454173 + 0.25

Combine like terms: -2.839454173 + 0.25 = -2.589454173
x = -2.589454173

Simplifying
x = -2.589454173


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.089454173, -2.589454173}
Solution
x = {3.089454173, -2.589454173}

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