9x^2-54x+69=0

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Solution for 9x^2-54x+69=0 equation: 


Simplifying
9x2 + -54x + 69 = 0

Reorder the terms:
69 + -54x + 9x2 = 0

Solving
69 + -54x + 9x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(23 + -18x + 3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(23 + -18x + 3x2)' equal to zero and attempt to solve:

Simplifying
23 + -18x + 3x2 = 0

Solving
23 + -18x + 3x2 = 0

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

Divide each side by '3'.
7.666666667 + -6x + x2 = 0

Move the constant term to the right:

Add '-7.666666667' to each side of the equation.
7.666666667 + -6x + -7.666666667 + x2 = 0 + -7.666666667

Reorder the terms:
7.666666667 + -7.666666667 + -6x + x2 = 0 + -7.666666667

Combine like terms: 7.666666667 + -7.666666667 = 0.000000000
0.000000000 + -6x + x2 = 0 + -7.666666667
-6x + x2 = 0 + -7.666666667

Combine like terms: 0 + -7.666666667 = -7.666666667
-6x + x2 = -7.666666667

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = -7.666666667 + 9

Reorder the terms:
9 + -6x + x2 = -7.666666667 + 9

Combine like terms: -7.666666667 + 9 = 1.333333333
9 + -6x + x2 = 1.333333333

Factor a perfect square on the left side:
(x + -3)(x + -3) = 1.333333333

Calculate the square root of the right side: 1.154700538

Break this problem into two subproblems by setting 
(x + -3) equal to 1.154700538 and -1.154700538.


Subproblem 1
x + -3 = 1.154700538

Simplifying
x + -3 = 1.154700538

Reorder the terms:
-3 + x = 1.154700538

Solving
-3 + x = 1.154700538

Solving for variable 'x'.

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

Add '3' to each side of the equation.
-3 + 3 + x = 1.154700538 + 3

Combine like terms: -3 + 3 = 0
0 + x = 1.154700538 + 3
x = 1.154700538 + 3

Combine like terms: 1.154700538 + 3 = 4.154700538
x = 4.154700538

Simplifying
x = 4.154700538


Subproblem 2
x + -3 = -1.154700538

Simplifying
x + -3 = -1.154700538

Reorder the terms:
-3 + x = -1.154700538

Solving
-3 + x = -1.154700538

Solving for variable 'x'.

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

Add '3' to each side of the equation.
-3 + 3 + x = -1.154700538 + 3

Combine like terms: -3 + 3 = 0
0 + x = -1.154700538 + 3
x = -1.154700538 + 3

Combine like terms: -1.154700538 + 3 = 1.845299462
x = 1.845299462

Simplifying
x = 1.845299462


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.154700538, 1.845299462}
Solution
x = {4.154700538, 1.845299462}

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