3x^2-24x-3=0

Solution for 3x^2-24x-3=0 equation:

Simplifying
3x2 + -24x + -3 = 0

Reorder the terms:
-3 + -24x + 3x2 = 0

Solving
-3 + -24x + 3x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + -8x + x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-1 + -8x + x2)' equal to zero and attempt to solve:

Simplifying
-1 + -8x + x2 = 0

Solving
-1 + -8x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -8x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -8x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -8x + x2 = 0 + 1
-8x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-8x + x2 = 1

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = 1 + 16

Reorder the terms:
16 + -8x + x2 = 1 + 16

Combine like terms: 1 + 16 = 17
16 + -8x + x2 = 17

Factor a perfect square on the left side:
(x + -4)(x + -4) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(x + -4) equal to 4.123105626 and -4.123105626.

Subproblem 1
x + -4 = 4.123105626

Simplifying
x + -4 = 4.123105626

Reorder the terms:
-4 + x = 4.123105626

Solving
-4 + x = 4.123105626

Solving for variable 'x'.

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

Add '4' to each side of the equation.
-4 + 4 + x = 4.123105626 + 4

Combine like terms: -4 + 4 = 0
0 + x = 4.123105626 + 4
x = 4.123105626 + 4

Combine like terms: 4.123105626 + 4 = 8.123105626
x = 8.123105626

Simplifying
x = 8.123105626

Subproblem 2
x + -4 = -4.123105626

Simplifying
x + -4 = -4.123105626

Reorder the terms:
-4 + x = -4.123105626

Solving
-4 + x = -4.123105626

Solving for variable 'x'.

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

Add '4' to each side of the equation.
-4 + 4 + x = -4.123105626 + 4

Combine like terms: -4 + 4 = 0
0 + x = -4.123105626 + 4
x = -4.123105626 + 4

Combine like terms: -4.123105626 + 4 = -0.123105626
x = -0.123105626

Simplifying
x = -0.123105626

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.123105626, -0.123105626}
Solution
x = {8.123105626, -0.123105626}