3f(4f-6)78=428

Solution for 3f(4f-6)78=428 equation:

Simplifying
3f(4f + -6) * 78 = 428

Reorder the terms:
3f(-6 + 4f) * 78 = 428

Reorder the terms for easier multiplication:
3 * 78f(-6 + 4f) = 428

Multiply 3 * 78
234f(-6 + 4f) = 428
(-6 * 234f + 4f * 234f) = 428
(-1404f + 936f2) = 428

Solving
-1404f + 936f2 = 428

Solving for variable 'f'.

Reorder the terms:
-428 + -1404f + 936f2 = 428 + -428

Combine like terms: 428 + -428 = 0
-428 + -1404f + 936f2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-107 + -351f + 234f2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-107 + -351f + 234f2)' equal to zero and attempt to solve:

Simplifying
-107 + -351f + 234f2 = 0

Solving
-107 + -351f + 234f2 = 0

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

Divide each side by '234'.
-0.4572649573 + -1.5f + f2 = 0

Move the constant term to the right:

Add '0.4572649573' to each side of the equation.
-0.4572649573 + -1.5f + 0.4572649573 + f2 = 0 + 0.4572649573

Reorder the terms:
-0.4572649573 + 0.4572649573 + -1.5f + f2 = 0 + 0.4572649573

Combine like terms: -0.4572649573 + 0.4572649573 = 0.0000000000
0.0000000000 + -1.5f + f2 = 0 + 0.4572649573
-1.5f + f2 = 0 + 0.4572649573

Combine like terms: 0 + 0.4572649573 = 0.4572649573
-1.5f + f2 = 0.4572649573

The f term is -1.5f.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5f + 0.5625 + f2 = 0.4572649573 + 0.5625

Reorder the terms:
0.5625 + -1.5f + f2 = 0.4572649573 + 0.5625

Combine like terms: 0.4572649573 + 0.5625 = 1.0197649573
0.5625 + -1.5f + f2 = 1.0197649573

Factor a perfect square on the left side:
(f + -0.75)(f + -0.75) = 1.0197649573

Calculate the square root of the right side: 1.009834124

Break this problem into two subproblems by setting 
(f + -0.75) equal to 1.009834124 and -1.009834124.

Subproblem 1
f + -0.75 = 1.009834124

Simplifying
f + -0.75 = 1.009834124

Reorder the terms:
-0.75 + f = 1.009834124

Solving
-0.75 + f = 1.009834124

Solving for variable 'f'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + f = 1.009834124 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + f = 1.009834124 + 0.75
f = 1.009834124 + 0.75

Combine like terms: 1.009834124 + 0.75 = 1.759834124
f = 1.759834124

Simplifying
f = 1.759834124

Subproblem 2
f + -0.75 = -1.009834124

Simplifying
f + -0.75 = -1.009834124

Reorder the terms:
-0.75 + f = -1.009834124

Solving
-0.75 + f = -1.009834124

Solving for variable 'f'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + f = -1.009834124 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + f = -1.009834124 + 0.75
f = -1.009834124 + 0.75

Combine like terms: -1.009834124 + 0.75 = -0.259834124
f = -0.259834124

Simplifying
f = -0.259834124

Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {1.759834124, -0.259834124}
Solution
f = {1.759834124, -0.259834124}