3x^2-22x+27=0

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Solution for 3x^2-22x+27=0 equation: 


Simplifying
3x2 + -22x + 27 = 0

Reorder the terms:
27 + -22x + 3x2 = 0

Solving
27 + -22x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
9 + -7.333333333x + x2 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -7.333333333x + -9 + x2 = 0 + -9

Reorder the terms:
9 + -9 + -7.333333333x + x2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -7.333333333x + x2 = 0 + -9
-7.333333333x + x2 = 0 + -9

Combine like terms: 0 + -9 = -9
-7.333333333x + x2 = -9

The x term is -7.333333333x.  Take half its coefficient (-3.666666667).
Square it (13.44444445) and add it to both sides.

Add '13.44444445' to each side of the equation.
-7.333333333x + 13.44444445 + x2 = -9 + 13.44444445

Reorder the terms:
13.44444445 + -7.333333333x + x2 = -9 + 13.44444445

Combine like terms: -9 + 13.44444445 = 4.44444445
13.44444445 + -7.333333333x + x2 = 4.44444445

Factor a perfect square on the left side:
(x + -3.666666667)(x + -3.666666667) = 4.44444445

Calculate the square root of the right side: 2.108185108

Break this problem into two subproblems by setting 
(x + -3.666666667) equal to 2.108185108 and -2.108185108.


Subproblem 1
x + -3.666666667 = 2.108185108

Simplifying
x + -3.666666667 = 2.108185108

Reorder the terms:
-3.666666667 + x = 2.108185108

Solving
-3.666666667 + x = 2.108185108

Solving for variable 'x'.

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

Add '3.666666667' to each side of the equation.
-3.666666667 + 3.666666667 + x = 2.108185108 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + x = 2.108185108 + 3.666666667
x = 2.108185108 + 3.666666667

Combine like terms: 2.108185108 + 3.666666667 = 5.774851775
x = 5.774851775

Simplifying
x = 5.774851775


Subproblem 2
x + -3.666666667 = -2.108185108

Simplifying
x + -3.666666667 = -2.108185108

Reorder the terms:
-3.666666667 + x = -2.108185108

Solving
-3.666666667 + x = -2.108185108

Solving for variable 'x'.

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

Add '3.666666667' to each side of the equation.
-3.666666667 + 3.666666667 + x = -2.108185108 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + x = -2.108185108 + 3.666666667
x = -2.108185108 + 3.666666667

Combine like terms: -2.108185108 + 3.666666667 = 1.558481559
x = 1.558481559

Simplifying
x = 1.558481559


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.774851775, 1.558481559}

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