6x^2+9x-27=27

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Solution for 6x^2+9x-27=27 equation: 


Simplifying
6x2 + 9x + -27 = 27

Reorder the terms:
-27 + 9x + 6x2 = 27

Solving
-27 + 9x + 6x2 = 27

Solving for variable 'x'.

Reorder the terms:
-27 + -27 + 9x + 6x2 = 27 + -27

Combine like terms: -27 + -27 = -54
-54 + 9x + 6x2 = 27 + -27

Combine like terms: 27 + -27 = 0
-54 + 9x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-18 + 3x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-18 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-18 + 3x + 2x2 = 0

Solving
-18 + 3x + 2x2 = 0

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

Divide each side by '2'.
-9 + 1.5x + x2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 1.5x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + 1.5x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 1.5x + x2 = 0 + 9
1.5x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
1.5x + x2 = 9

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 = 9 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 9 + 0.5625

Combine like terms: 9 + 0.5625 = 9.5625
0.5625 + 1.5x + x2 = 9.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(x + 0.75) equal to 3.092329219 and -3.092329219.


Subproblem 1
x + 0.75 = 3.092329219

Simplifying
x + 0.75 = 3.092329219

Reorder the terms:
0.75 + x = 3.092329219

Solving
0.75 + x = 3.092329219

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 = 3.092329219 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 3.092329219 + -0.75
x = 3.092329219 + -0.75

Combine like terms: 3.092329219 + -0.75 = 2.342329219
x = 2.342329219

Simplifying
x = 2.342329219


Subproblem 2
x + 0.75 = -3.092329219

Simplifying
x + 0.75 = -3.092329219

Reorder the terms:
0.75 + x = -3.092329219

Solving
0.75 + x = -3.092329219

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 = -3.092329219 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -3.092329219 + -0.75
x = -3.092329219 + -0.75

Combine like terms: -3.092329219 + -0.75 = -3.842329219
x = -3.842329219

Simplifying
x = -3.842329219


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.342329219, -3.842329219}
Solution
x = {2.342329219, -3.842329219}

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