24x^2-6x-1=0

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Solution for 24x^2-6x-1=0 equation: 


Simplifying
24x2 + -6x + -1 = 0

Reorder the terms:
-1 + -6x + 24x2 = 0

Solving
-1 + -6x + 24x2 = 0

Solving for variable 'x'.

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

Divide each side by '24'.
-0.04166666667 + -0.25x + x2 = 0

Move the constant term to the right:

Add '0.04166666667' to each side of the equation.
-0.04166666667 + -0.25x + 0.04166666667 + x2 = 0 + 0.04166666667

Reorder the terms:
-0.04166666667 + 0.04166666667 + -0.25x + x2 = 0 + 0.04166666667

Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000
0.00000000000 + -0.25x + x2 = 0 + 0.04166666667
-0.25x + x2 = 0 + 0.04166666667

Combine like terms: 0 + 0.04166666667 = 0.04166666667
-0.25x + x2 = 0.04166666667

The x term is -0.25x.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25x + 0.015625 + x2 = 0.04166666667 + 0.015625

Reorder the terms:
0.015625 + -0.25x + x2 = 0.04166666667 + 0.015625

Combine like terms: 0.04166666667 + 0.015625 = 0.05729166667
0.015625 + -0.25x + x2 = 0.05729166667

Factor a perfect square on the left side:
(x + -0.125)(x + -0.125) = 0.05729166667

Calculate the square root of the right side: 0.239356777

Break this problem into two subproblems by setting 
(x + -0.125) equal to 0.239356777 and -0.239356777.


Subproblem 1
x + -0.125 = 0.239356777

Simplifying
x + -0.125 = 0.239356777

Reorder the terms:
-0.125 + x = 0.239356777

Solving
-0.125 + x = 0.239356777

Solving for variable 'x'.

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

Add '0.125' to each side of the equation.
-0.125 + 0.125 + x = 0.239356777 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + x = 0.239356777 + 0.125
x = 0.239356777 + 0.125

Combine like terms: 0.239356777 + 0.125 = 0.364356777
x = 0.364356777

Simplifying
x = 0.364356777


Subproblem 2
x + -0.125 = -0.239356777

Simplifying
x + -0.125 = -0.239356777

Reorder the terms:
-0.125 + x = -0.239356777

Solving
-0.125 + x = -0.239356777

Solving for variable 'x'.

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

Add '0.125' to each side of the equation.
-0.125 + 0.125 + x = -0.239356777 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + x = -0.239356777 + 0.125
x = -0.239356777 + 0.125

Combine like terms: -0.239356777 + 0.125 = -0.114356777
x = -0.114356777

Simplifying
x = -0.114356777


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.364356777, -0.114356777}

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