-6x^2-9x+3=0

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Solution for -6x^2-9x+3=0 equation: 


Simplifying
-6x2 + -9x + 3 = 0

Reorder the terms:
3 + -9x + -6x2 = 0

Solving
3 + -9x + -6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(1 + -3x + -2x2) = 0

Ignore the factor 3.

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

Simplifying
1 + -3x + -2x2 = 0

Solving
1 + -3x + -2x2 = 0

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

Divide each side by '-2'.
-0.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 1.5x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 1.5x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 1.5x + x2 = 0 + 0.5
1.5x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
1.5x + x2 = 0.5

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 = 0.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 0.5 + 0.5625

Combine like terms: 0.5 + 0.5625 = 1.0625
0.5625 + 1.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + 0.75) equal to 1.030776406 and -1.030776406.


Subproblem 1
x + 0.75 = 1.030776406

Simplifying
x + 0.75 = 1.030776406

Reorder the terms:
0.75 + x = 1.030776406

Solving
0.75 + x = 1.030776406

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 = 1.030776406 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 1.030776406 + -0.75
x = 1.030776406 + -0.75

Combine like terms: 1.030776406 + -0.75 = 0.280776406
x = 0.280776406

Simplifying
x = 0.280776406


Subproblem 2
x + 0.75 = -1.030776406

Simplifying
x + 0.75 = -1.030776406

Reorder the terms:
0.75 + x = -1.030776406

Solving
0.75 + x = -1.030776406

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 = -1.030776406 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -1.030776406 + -0.75
x = -1.030776406 + -0.75

Combine like terms: -1.030776406 + -0.75 = -1.780776406
x = -1.780776406

Simplifying
x = -1.780776406


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.280776406, -1.780776406}
Solution
x = {0.280776406, -1.780776406}

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