12x^2+27x-12=0

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


Simplifying
12x2 + 27x + -12 = 0

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

Solving
-12 + 27x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-4 + 9x + 4x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-4 + 9x + 4x2)' equal to zero and attempt to solve:

Simplifying
-4 + 9x + 4x2 = 0

Solving
-4 + 9x + 4x2 = 0

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

Divide each side by '4'.
-1 + 2.25x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2.25x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 2.25x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2.25x + x2 = 0 + 1
2.25x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
2.25x + x2 = 1

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = 1 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = 1 + 1.265625

Combine like terms: 1 + 1.265625 = 2.265625
1.265625 + 2.25x + x2 = 2.265625

Factor a perfect square on the left side:
(x + 1.125)(x + 1.125) = 2.265625

Calculate the square root of the right side: 1.505199322

Break this problem into two subproblems by setting 
(x + 1.125) equal to 1.505199322 and -1.505199322.


Subproblem 1
x + 1.125 = 1.505199322

Simplifying
x + 1.125 = 1.505199322

Reorder the terms:
1.125 + x = 1.505199322

Solving
1.125 + x = 1.505199322

Solving for variable 'x'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = 1.505199322 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = 1.505199322 + -1.125
x = 1.505199322 + -1.125

Combine like terms: 1.505199322 + -1.125 = 0.380199322
x = 0.380199322

Simplifying
x = 0.380199322


Subproblem 2
x + 1.125 = -1.505199322

Simplifying
x + 1.125 = -1.505199322

Reorder the terms:
1.125 + x = -1.505199322

Solving
1.125 + x = -1.505199322

Solving for variable 'x'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = -1.505199322 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = -1.505199322 + -1.125
x = -1.505199322 + -1.125

Combine like terms: -1.505199322 + -1.125 = -2.630199322
x = -2.630199322

Simplifying
x = -2.630199322


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.380199322, -2.630199322}
Solution
x = {0.380199322, -2.630199322}

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