4f^2-9f+3=0

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Solution for 4f^2-9f+3=0 equation: 


Simplifying
4f2 + -9f + 3 = 0

Reorder the terms:
3 + -9f + 4f2 = 0

Solving
3 + -9f + 4f2 = 0

Solving for variable 'f'.

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

Divide each side by '4'.
0.75 + -2.25f + f2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + -2.25f + -0.75 + f2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + -2.25f + f2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + -2.25f + f2 = 0 + -0.75
-2.25f + f2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
-2.25f + f2 = -0.75

The f term is -2.25f.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25f + 1.265625 + f2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + -2.25f + f2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + -2.25f + f2 = 0.515625

Factor a perfect square on the left side:
(f + -1.125)(f + -1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(f + -1.125) equal to 0.718070331 and -0.718070331.


Subproblem 1
f + -1.125 = 0.718070331

Simplifying
f + -1.125 = 0.718070331

Reorder the terms:
-1.125 + f = 0.718070331

Solving
-1.125 + f = 0.718070331

Solving for variable 'f'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + f = 0.718070331 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + f = 0.718070331 + 1.125
f = 0.718070331 + 1.125

Combine like terms: 0.718070331 + 1.125 = 1.843070331
f = 1.843070331

Simplifying
f = 1.843070331


Subproblem 2
f + -1.125 = -0.718070331

Simplifying
f + -1.125 = -0.718070331

Reorder the terms:
-1.125 + f = -0.718070331

Solving
-1.125 + f = -0.718070331

Solving for variable 'f'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + f = -0.718070331 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + f = -0.718070331 + 1.125
f = -0.718070331 + 1.125

Combine like terms: -0.718070331 + 1.125 = 0.406929669
f = 0.406929669

Simplifying
f = 0.406929669


Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {1.843070331, 0.406929669}

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