2f(5f-2)-10(f+f-3f+6)=-8f(f+4)+4(2f+f-7f)

Solution for 2f(5f-2)-10(f+f-3f+6)=-8f(f+4)+4(2f+f-7f) equation:

Simplifying
2f(5f + -2) + -10(f + f + -3f + 6) = -8f(f + 4) + 4(2f + f + -7f)

Reorder the terms:
2f(-2 + 5f) + -10(f + f + -3f + 6) = -8f(f + 4) + 4(2f + f + -7f)
(-2 * 2f + 5f * 2f) + -10(f + f + -3f + 6) = -8f(f + 4) + 4(2f + f + -7f)
(-4f + 10f2) + -10(f + f + -3f + 6) = -8f(f + 4) + 4(2f + f + -7f)

Reorder the terms:
-4f + 10f2 + -10(6 + f + f + -3f) = -8f(f + 4) + 4(2f + f + -7f)

Combine like terms: f + f = 2f
-4f + 10f2 + -10(6 + 2f + -3f) = -8f(f + 4) + 4(2f + f + -7f)

Combine like terms: 2f + -3f = -1f
-4f + 10f2 + -10(6 + -1f) = -8f(f + 4) + 4(2f + f + -7f)
-4f + 10f2 + (6 * -10 + -1f * -10) = -8f(f + 4) + 4(2f + f + -7f)
-4f + 10f2 + (-60 + 10f) = -8f(f + 4) + 4(2f + f + -7f)

Reorder the terms:
-60 + -4f + 10f + 10f2 = -8f(f + 4) + 4(2f + f + -7f)

Combine like terms: -4f + 10f = 6f
-60 + 6f + 10f2 = -8f(f + 4) + 4(2f + f + -7f)

Reorder the terms:
-60 + 6f + 10f2 = -8f(4 + f) + 4(2f + f + -7f)
-60 + 6f + 10f2 = (4 * -8f + f * -8f) + 4(2f + f + -7f)
-60 + 6f + 10f2 = (-32f + -8f2) + 4(2f + f + -7f)

Combine like terms: 2f + f = 3f
-60 + 6f + 10f2 = -32f + -8f2 + 4(3f + -7f)

Combine like terms: 3f + -7f = -4f
-60 + 6f + 10f2 = -32f + -8f2 + 4(-4f)

Remove parenthesis around (-4f)
-60 + 6f + 10f2 = -32f + -8f2 + 4 * -4f

Multiply 4 * -4
-60 + 6f + 10f2 = -32f + -8f2 + -16f

Reorder the terms:
-60 + 6f + 10f2 = -32f + -16f + -8f2

Combine like terms: -32f + -16f = -48f
-60 + 6f + 10f2 = -48f + -8f2

Solving
-60 + 6f + 10f2 = -48f + -8f2

Solving for variable 'f'.

Reorder the terms:
-60 + 6f + 48f + 10f2 + 8f2 = -48f + -8f2 + 48f + 8f2

Combine like terms: 6f + 48f = 54f
-60 + 54f + 10f2 + 8f2 = -48f + -8f2 + 48f + 8f2

Combine like terms: 10f2 + 8f2 = 18f2
-60 + 54f + 18f2 = -48f + -8f2 + 48f + 8f2

Reorder the terms:
-60 + 54f + 18f2 = -48f + 48f + -8f2 + 8f2

Combine like terms: -48f + 48f = 0
-60 + 54f + 18f2 = 0 + -8f2 + 8f2
-60 + 54f + 18f2 = -8f2 + 8f2

Combine like terms: -8f2 + 8f2 = 0
-60 + 54f + 18f2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-10 + 9f + 3f2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(-10 + 9f + 3f2)' equal to zero and attempt to solve:

Simplifying
-10 + 9f + 3f2 = 0

Solving
-10 + 9f + 3f2 = 0

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

Divide each side by '3'.
-3.333333333 + 3f + f2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + 3f + 3.333333333 + f2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + 3f + f2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + 3f + f2 = 0 + 3.333333333
3f + f2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
3f + f2 = 3.333333333

The f term is 3f.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3f + 2.25 + f2 = 3.333333333 + 2.25

Reorder the terms:
2.25 + 3f + f2 = 3.333333333 + 2.25

Combine like terms: 3.333333333 + 2.25 = 5.583333333
2.25 + 3f + f2 = 5.583333333

Factor a perfect square on the left side:
(f + 1.5)(f + 1.5) = 5.583333333

Calculate the square root of the right side: 2.362907813

Break this problem into two subproblems by setting 
(f + 1.5) equal to 2.362907813 and -2.362907813.

Subproblem 1
f + 1.5 = 2.362907813

Simplifying
f + 1.5 = 2.362907813

Reorder the terms:
1.5 + f = 2.362907813

Solving
1.5 + f = 2.362907813

Solving for variable 'f'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + f = 2.362907813 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + f = 2.362907813 + -1.5
f = 2.362907813 + -1.5

Combine like terms: 2.362907813 + -1.5 = 0.862907813
f = 0.862907813

Simplifying
f = 0.862907813

Subproblem 2
f + 1.5 = -2.362907813

Simplifying
f + 1.5 = -2.362907813

Reorder the terms:
1.5 + f = -2.362907813

Solving
1.5 + f = -2.362907813

Solving for variable 'f'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + f = -2.362907813 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + f = -2.362907813 + -1.5
f = -2.362907813 + -1.5

Combine like terms: -2.362907813 + -1.5 = -3.862907813
f = -3.862907813

Simplifying
f = -3.862907813

Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {0.862907813, -3.862907813}
Solution
f = {0.862907813, -3.862907813}