8x^2-41x-5=0

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Solution for 8x^2-41x-5=0 equation: 


Simplifying
8x2 + -41x + -5 = 0

Reorder the terms:
-5 + -41x + 8x2 = 0

Solving
-5 + -41x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
-0.625 + -5.125x + x2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + -5.125x + 0.625 + x2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + -5.125x + x2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + -5.125x + x2 = 0 + 0.625
-5.125x + x2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
-5.125x + x2 = 0.625

The x term is -5.125x.  Take half its coefficient (-2.5625).
Square it (6.56640625) and add it to both sides.

Add '6.56640625' to each side of the equation.
-5.125x + 6.56640625 + x2 = 0.625 + 6.56640625

Reorder the terms:
6.56640625 + -5.125x + x2 = 0.625 + 6.56640625

Combine like terms: 0.625 + 6.56640625 = 7.19140625
6.56640625 + -5.125x + x2 = 7.19140625

Factor a perfect square on the left side:
(x + -2.5625)(x + -2.5625) = 7.19140625

Calculate the square root of the right side: 2.681679744

Break this problem into two subproblems by setting 
(x + -2.5625) equal to 2.681679744 and -2.681679744.


Subproblem 1
x + -2.5625 = 2.681679744

Simplifying
x + -2.5625 = 2.681679744

Reorder the terms:
-2.5625 + x = 2.681679744

Solving
-2.5625 + x = 2.681679744

Solving for variable 'x'.

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

Add '2.5625' to each side of the equation.
-2.5625 + 2.5625 + x = 2.681679744 + 2.5625

Combine like terms: -2.5625 + 2.5625 = 0.0000
0.0000 + x = 2.681679744 + 2.5625
x = 2.681679744 + 2.5625

Combine like terms: 2.681679744 + 2.5625 = 5.244179744
x = 5.244179744

Simplifying
x = 5.244179744


Subproblem 2
x + -2.5625 = -2.681679744

Simplifying
x + -2.5625 = -2.681679744

Reorder the terms:
-2.5625 + x = -2.681679744

Solving
-2.5625 + x = -2.681679744

Solving for variable 'x'.

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

Add '2.5625' to each side of the equation.
-2.5625 + 2.5625 + x = -2.681679744 + 2.5625

Combine like terms: -2.5625 + 2.5625 = 0.0000
0.0000 + x = -2.681679744 + 2.5625
x = -2.681679744 + 2.5625

Combine like terms: -2.681679744 + 2.5625 = -0.119179744
x = -0.119179744

Simplifying
x = -0.119179744


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.244179744, -0.119179744}

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