8x^2-41x-65=0

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Solution for 8x^2-41x-65=0 equation:


Simplifying
8x2 + -41x + -65 = 0

Reorder the terms:
-65 + -41x + 8x2 = 0

Solving
-65 + -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'.
-8.125 + -5.125x + x2 = 0

Move the constant term to the right:

Add '8.125' to each side of the equation.
-8.125 + -5.125x + 8.125 + x2 = 0 + 8.125

Reorder the terms:
-8.125 + 8.125 + -5.125x + x2 = 0 + 8.125

Combine like terms: -8.125 + 8.125 = 0.000
0.000 + -5.125x + x2 = 0 + 8.125
-5.125x + x2 = 0 + 8.125

Combine like terms: 0 + 8.125 = 8.125
-5.125x + x2 = 8.125

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 = 8.125 + 6.56640625

Reorder the terms:
6.56640625 + -5.125x + x2 = 8.125 + 6.56640625

Combine like terms: 8.125 + 6.56640625 = 14.69140625
6.56640625 + -5.125x + x2 = 14.69140625

Factor a perfect square on the left side:
(x + -2.5625)(x + -2.5625) = 14.69140625

Calculate the square root of the right side: 3.832937027

Break this problem into two subproblems by setting 
(x + -2.5625) equal to 3.832937027 and -3.832937027.

Subproblem 1

x + -2.5625 = 3.832937027 Simplifying x + -2.5625 = 3.832937027 Reorder the terms: -2.5625 + x = 3.832937027 Solving -2.5625 + x = 3.832937027 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 = 3.832937027 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + x = 3.832937027 + 2.5625 x = 3.832937027 + 2.5625 Combine like terms: 3.832937027 + 2.5625 = 6.395437027 x = 6.395437027 Simplifying x = 6.395437027

Subproblem 2

x + -2.5625 = -3.832937027 Simplifying x + -2.5625 = -3.832937027 Reorder the terms: -2.5625 + x = -3.832937027 Solving -2.5625 + x = -3.832937027 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 = -3.832937027 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + x = -3.832937027 + 2.5625 x = -3.832937027 + 2.5625 Combine like terms: -3.832937027 + 2.5625 = -1.270437027 x = -1.270437027 Simplifying x = -1.270437027

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.395437027, -1.270437027}

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