8x^2-12x-52=0

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Solution for 8x^2-12x-52=0 equation: 


Simplifying
8x2 + -12x + -52 = 0

Reorder the terms:
-52 + -12x + 8x2 = 0

Solving
-52 + -12x + 8x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-13 + -3x + 2x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-13 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-13 + -3x + 2x2 = 0

Solving
-13 + -3x + 2x2 = 0

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

Divide each side by '2'.
-6.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + -1.5x + 6.5 + x2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + -1.5x + x2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + -1.5x + x2 = 0 + 6.5
-1.5x + x2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
-1.5x + x2 = 6.5

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 6.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 6.5 + 0.5625

Combine like terms: 6.5 + 0.5625 = 7.0625
0.5625 + -1.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + -0.75) equal to 2.657536453 and -2.657536453.


Subproblem 1
x + -0.75 = 2.657536453

Simplifying
x + -0.75 = 2.657536453

Reorder the terms:
-0.75 + x = 2.657536453

Solving
-0.75 + x = 2.657536453

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 2.657536453 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 2.657536453 + 0.75
x = 2.657536453 + 0.75

Combine like terms: 2.657536453 + 0.75 = 3.407536453
x = 3.407536453

Simplifying
x = 3.407536453


Subproblem 2
x + -0.75 = -2.657536453

Simplifying
x + -0.75 = -2.657536453

Reorder the terms:
-0.75 + x = -2.657536453

Solving
-0.75 + x = -2.657536453

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -2.657536453 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -2.657536453 + 0.75
x = -2.657536453 + 0.75

Combine like terms: -2.657536453 + 0.75 = -1.907536453
x = -1.907536453

Simplifying
x = -1.907536453


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.407536453, -1.907536453}
Solution
x = {3.407536453, -1.907536453}

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