6x^2-9x-24=0

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Solution for 6x^2-9x-24=0 equation: 


Simplifying
6x2 + -9x + -24 = 0

Reorder the terms:
-24 + -9x + 6x2 = 0

Solving
-24 + -9x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-8 + -3x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-8 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-8 + -3x + 2x2 = 0

Solving
-8 + -3x + 2x2 = 0

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

Divide each side by '2'.
-4 + -1.5x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1.5x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -1.5x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1.5x + x2 = 0 + 4
-1.5x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-1.5x + x2 = 4

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 4 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 4 + 0.5625

Combine like terms: 4 + 0.5625 = 4.5625
0.5625 + -1.5x + x2 = 4.5625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(x + -0.75) equal to 2.136000936 and -2.136000936.


Subproblem 1
x + -0.75 = 2.136000936

Simplifying
x + -0.75 = 2.136000936

Reorder the terms:
-0.75 + x = 2.136000936

Solving
-0.75 + x = 2.136000936

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 2.136000936 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 2.136000936 + 0.75
x = 2.136000936 + 0.75

Combine like terms: 2.136000936 + 0.75 = 2.886000936
x = 2.886000936

Simplifying
x = 2.886000936


Subproblem 2
x + -0.75 = -2.136000936

Simplifying
x + -0.75 = -2.136000936

Reorder the terms:
-0.75 + x = -2.136000936

Solving
-0.75 + x = -2.136000936

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -2.136000936 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -2.136000936 + 0.75
x = -2.136000936 + 0.75

Combine like terms: -2.136000936 + 0.75 = -1.386000936
x = -1.386000936

Simplifying
x = -1.386000936


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.886000936, -1.386000936}
Solution
x = {2.886000936, -1.386000936}

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