9x^2+84x+135=0

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Solution for 9x^2+84x+135=0 equation: 


Simplifying
9x2 + 84x + 135 = 0

Reorder the terms:
135 + 84x + 9x2 = 0

Solving
135 + 84x + 9x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(45 + 28x + 3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(45 + 28x + 3x2)' equal to zero and attempt to solve:

Simplifying
45 + 28x + 3x2 = 0

Solving
45 + 28x + 3x2 = 0

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

Divide each side by '3'.
15 + 9.333333333x + x2 = 0

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + 9.333333333x + -15 + x2 = 0 + -15

Reorder the terms:
15 + -15 + 9.333333333x + x2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + 9.333333333x + x2 = 0 + -15
9.333333333x + x2 = 0 + -15

Combine like terms: 0 + -15 = -15
9.333333333x + x2 = -15

The x term is 9.333333333x.  Take half its coefficient (4.666666667).
Square it (21.77777778) and add it to both sides.

Add '21.77777778' to each side of the equation.
9.333333333x + 21.77777778 + x2 = -15 + 21.77777778

Reorder the terms:
21.77777778 + 9.333333333x + x2 = -15 + 21.77777778

Combine like terms: -15 + 21.77777778 = 6.77777778
21.77777778 + 9.333333333x + x2 = 6.77777778

Factor a perfect square on the left side:
(x + 4.666666667)(x + 4.666666667) = 6.77777778

Calculate the square root of the right side: 2.603416559

Break this problem into two subproblems by setting 
(x + 4.666666667) equal to 2.603416559 and -2.603416559.


Subproblem 1
x + 4.666666667 = 2.603416559

Simplifying
x + 4.666666667 = 2.603416559

Reorder the terms:
4.666666667 + x = 2.603416559

Solving
4.666666667 + x = 2.603416559

Solving for variable 'x'.

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

Add '-4.666666667' to each side of the equation.
4.666666667 + -4.666666667 + x = 2.603416559 + -4.666666667

Combine like terms: 4.666666667 + -4.666666667 = 0.000000000
0.000000000 + x = 2.603416559 + -4.666666667
x = 2.603416559 + -4.666666667

Combine like terms: 2.603416559 + -4.666666667 = -2.063250108
x = -2.063250108

Simplifying
x = -2.063250108


Subproblem 2
x + 4.666666667 = -2.603416559

Simplifying
x + 4.666666667 = -2.603416559

Reorder the terms:
4.666666667 + x = -2.603416559

Solving
4.666666667 + x = -2.603416559

Solving for variable 'x'.

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

Add '-4.666666667' to each side of the equation.
4.666666667 + -4.666666667 + x = -2.603416559 + -4.666666667

Combine like terms: 4.666666667 + -4.666666667 = 0.000000000
0.000000000 + x = -2.603416559 + -4.666666667
x = -2.603416559 + -4.666666667

Combine like terms: -2.603416559 + -4.666666667 = -7.270083226
x = -7.270083226

Simplifying
x = -7.270083226


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.063250108, -7.270083226}
Solution
x = {-2.063250108, -7.270083226}

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