12x^2-6x-27+7=0

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Solution for 12x^2-6x-27+7=0 equation: 


Simplifying
12x2 + -6x + -27 + 7 = 0

Reorder the terms:
-27 + 7 + -6x + 12x2 = 0

Combine like terms: -27 + 7 = -20
-20 + -6x + 12x2 = 0

Solving
-20 + -6x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-10 + -3x + 6x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-10 + -3x + 6x2)' equal to zero and attempt to solve:

Simplifying
-10 + -3x + 6x2 = 0

Solving
-10 + -3x + 6x2 = 0

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

Divide each side by '6'.
-1.666666667 + -0.5x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -0.5x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -0.5x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -0.5x + x2 = 0 + 1.666666667
-0.5x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-0.5x + x2 = 1.666666667

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 1.666666667 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1.666666667 + 0.0625

Combine like terms: 1.666666667 + 0.0625 = 1.729166667
0.0625 + -0.5x + x2 = 1.729166667

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.729166667

Calculate the square root of the right side: 1.31497782

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.31497782 and -1.31497782.


Subproblem 1
x + -0.25 = 1.31497782

Simplifying
x + -0.25 = 1.31497782

Reorder the terms:
-0.25 + x = 1.31497782

Solving
-0.25 + x = 1.31497782

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 1.31497782 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 1.31497782 + 0.25
x = 1.31497782 + 0.25

Combine like terms: 1.31497782 + 0.25 = 1.56497782
x = 1.56497782

Simplifying
x = 1.56497782


Subproblem 2
x + -0.25 = -1.31497782

Simplifying
x + -0.25 = -1.31497782

Reorder the terms:
-0.25 + x = -1.31497782

Solving
-0.25 + x = -1.31497782

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -1.31497782 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -1.31497782 + 0.25
x = -1.31497782 + 0.25

Combine like terms: -1.31497782 + 0.25 = -1.06497782
x = -1.06497782

Simplifying
x = -1.06497782


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.56497782, -1.06497782}
Solution
x = {1.56497782, -1.06497782}

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