6x^2-30x-41=0

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Solution for 6x^2-30x-41=0 equation: 


Simplifying
6x2 + -30x + -41 = 0

Reorder the terms:
-41 + -30x + 6x2 = 0

Solving
-41 + -30x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-6.833333333 + -5x + x2 = 0

Move the constant term to the right:

Add '6.833333333' to each side of the equation.
-6.833333333 + -5x + 6.833333333 + x2 = 0 + 6.833333333

Reorder the terms:
-6.833333333 + 6.833333333 + -5x + x2 = 0 + 6.833333333

Combine like terms: -6.833333333 + 6.833333333 = 0.000000000
0.000000000 + -5x + x2 = 0 + 6.833333333
-5x + x2 = 0 + 6.833333333

Combine like terms: 0 + 6.833333333 = 6.833333333
-5x + x2 = 6.833333333

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 6.833333333 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 6.833333333 + 6.25

Combine like terms: 6.833333333 + 6.25 = 13.083333333
6.25 + -5x + x2 = 13.083333333

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 13.083333333

Calculate the square root of the right side: 3.617089069

Break this problem into two subproblems by setting 
(x + -2.5) equal to 3.617089069 and -3.617089069.


Subproblem 1
x + -2.5 = 3.617089069

Simplifying
x + -2.5 = 3.617089069

Reorder the terms:
-2.5 + x = 3.617089069

Solving
-2.5 + x = 3.617089069

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 3.617089069 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 3.617089069 + 2.5
x = 3.617089069 + 2.5

Combine like terms: 3.617089069 + 2.5 = 6.117089069
x = 6.117089069

Simplifying
x = 6.117089069


Subproblem 2
x + -2.5 = -3.617089069

Simplifying
x + -2.5 = -3.617089069

Reorder the terms:
-2.5 + x = -3.617089069

Solving
-2.5 + x = -3.617089069

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -3.617089069 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -3.617089069 + 2.5
x = -3.617089069 + 2.5

Combine like terms: -3.617089069 + 2.5 = -1.117089069
x = -1.117089069

Simplifying
x = -1.117089069


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.117089069, -1.117089069}

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