3f^2-66f+27=0

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Solution for 3f^2-66f+27=0 equation: 


Simplifying
3f2 + -66f + 27 = 0

Reorder the terms:
27 + -66f + 3f2 = 0

Solving
27 + -66f + 3f2 = 0

Solving for variable 'f'.

Factor out the Greatest Common Factor (GCF), '3'.
3(9 + -22f + f2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(9 + -22f + f2)' equal to zero and attempt to solve:

Simplifying
9 + -22f + f2 = 0

Solving
9 + -22f + f2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -22f + -9 + f2 = 0 + -9

Reorder the terms:
9 + -9 + -22f + f2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -22f + f2 = 0 + -9
-22f + f2 = 0 + -9

Combine like terms: 0 + -9 = -9
-22f + f2 = -9

The f term is -22f.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22f + 121 + f2 = -9 + 121

Reorder the terms:
121 + -22f + f2 = -9 + 121

Combine like terms: -9 + 121 = 112
121 + -22f + f2 = 112

Factor a perfect square on the left side:
(f + -11)(f + -11) = 112

Calculate the square root of the right side: 10.583005244

Break this problem into two subproblems by setting 
(f + -11) equal to 10.583005244 and -10.583005244.


Subproblem 1
f + -11 = 10.583005244

Simplifying
f + -11 = 10.583005244

Reorder the terms:
-11 + f = 10.583005244

Solving
-11 + f = 10.583005244

Solving for variable 'f'.

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

Add '11' to each side of the equation.
-11 + 11 + f = 10.583005244 + 11

Combine like terms: -11 + 11 = 0
0 + f = 10.583005244 + 11
f = 10.583005244 + 11

Combine like terms: 10.583005244 + 11 = 21.583005244
f = 21.583005244

Simplifying
f = 21.583005244


Subproblem 2
f + -11 = -10.583005244

Simplifying
f + -11 = -10.583005244

Reorder the terms:
-11 + f = -10.583005244

Solving
-11 + f = -10.583005244

Solving for variable 'f'.

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

Add '11' to each side of the equation.
-11 + 11 + f = -10.583005244 + 11

Combine like terms: -11 + 11 = 0
0 + f = -10.583005244 + 11
f = -10.583005244 + 11

Combine like terms: -10.583005244 + 11 = 0.416994756
f = 0.416994756

Simplifying
f = 0.416994756


Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {21.583005244, 0.416994756}
Solution
f = {21.583005244, 0.416994756}

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