8x^2-6x-32=0

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Solution for 8x^2-6x-32=0 equation: 


Simplifying
8x2 + -6x + -32 = 0

Reorder the terms:
-32 + -6x + 8x2 = 0

Solving
-32 + -6x + 8x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-16 + -3x + 4x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-16 + -3x + 4x2)' equal to zero and attempt to solve:

Simplifying
-16 + -3x + 4x2 = 0

Solving
-16 + -3x + 4x2 = 0

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

Divide each side by '4'.
-4 + -0.75x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -0.75x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -0.75x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -0.75x + x2 = 0 + 4
-0.75x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-0.75x + x2 = 4

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = 4 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 4 + 0.140625

Combine like terms: 4 + 0.140625 = 4.140625
0.140625 + -0.75x + x2 = 4.140625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 4.140625

Calculate the square root of the right side: 2.034852575

Break this problem into two subproblems by setting 
(x + -0.375) equal to 2.034852575 and -2.034852575.


Subproblem 1
x + -0.375 = 2.034852575

Simplifying
x + -0.375 = 2.034852575

Reorder the terms:
-0.375 + x = 2.034852575

Solving
-0.375 + x = 2.034852575

Solving for variable 'x'.

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

Add '0.375' to each side of the equation.
-0.375 + 0.375 + x = 2.034852575 + 0.375

Combine like terms: -0.375 + 0.375 = 0.000
0.000 + x = 2.034852575 + 0.375
x = 2.034852575 + 0.375

Combine like terms: 2.034852575 + 0.375 = 2.409852575
x = 2.409852575

Simplifying
x = 2.409852575


Subproblem 2
x + -0.375 = -2.034852575

Simplifying
x + -0.375 = -2.034852575

Reorder the terms:
-0.375 + x = -2.034852575

Solving
-0.375 + x = -2.034852575

Solving for variable 'x'.

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

Add '0.375' to each side of the equation.
-0.375 + 0.375 + x = -2.034852575 + 0.375

Combine like terms: -0.375 + 0.375 = 0.000
0.000 + x = -2.034852575 + 0.375
x = -2.034852575 + 0.375

Combine like terms: -2.034852575 + 0.375 = -1.659852575
x = -1.659852575

Simplifying
x = -1.659852575


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.409852575, -1.659852575}
Solution
x = {2.409852575, -1.659852575}

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