2f(3f+2)=3

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Solution for 2f(3f+2)=3 equation:


Simplifying
2f(3f + 2) = 3

Reorder the terms:
2f(2 + 3f) = 3
(2 * 2f + 3f * 2f) = 3
(4f + 6f2) = 3

Solving
4f + 6f2 = 3

Solving for variable 'f'.

Reorder the terms:
-3 + 4f + 6f2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + 4f + 6f2 = 0

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

Divide each side by '6'.
-0.5 + 0.6666666667f + f2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.6666666667f + 0.5 + f2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.6666666667f + f2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.6666666667f + f2 = 0 + 0.5
0.6666666667f + f2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.6666666667f + f2 = 0.5

The f term is 0.6666666667f.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667f + 0.1111111112 + f2 = 0.5 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667f + f2 = 0.5 + 0.1111111112

Combine like terms: 0.5 + 0.1111111112 = 0.6111111112
0.1111111112 + 0.6666666667f + f2 = 0.6111111112

Factor a perfect square on the left side:
(f + 0.3333333334)(f + 0.3333333334) = 0.6111111112

Calculate the square root of the right side: 0.78173596

Break this problem into two subproblems by setting 
(f + 0.3333333334) equal to 0.78173596 and -0.78173596.

Subproblem 1

f + 0.3333333334 = 0.78173596 Simplifying f + 0.3333333334 = 0.78173596 Reorder the terms: 0.3333333334 + f = 0.78173596 Solving 0.3333333334 + f = 0.78173596 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + f = 0.78173596 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + f = 0.78173596 + -0.3333333334 f = 0.78173596 + -0.3333333334 Combine like terms: 0.78173596 + -0.3333333334 = 0.4484026266 f = 0.4484026266 Simplifying f = 0.4484026266

Subproblem 2

f + 0.3333333334 = -0.78173596 Simplifying f + 0.3333333334 = -0.78173596 Reorder the terms: 0.3333333334 + f = -0.78173596 Solving 0.3333333334 + f = -0.78173596 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + f = -0.78173596 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + f = -0.78173596 + -0.3333333334 f = -0.78173596 + -0.3333333334 Combine like terms: -0.78173596 + -0.3333333334 = -1.1150692934 f = -1.1150692934 Simplifying f = -1.1150692934

Solution

The solution to the problem is based on the solutions from the subproblems. f = {0.4484026266, -1.1150692934}

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