f^2-20f-41=0

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Solution for f^2-20f-41=0 equation: 


Simplifying
f2 + -20f + -41 = 0

Reorder the terms:
-41 + -20f + f2 = 0

Solving
-41 + -20f + f2 = 0

Solving for variable 'f'.

Begin completing the square.

Move the constant term to the right:

Add '41' to each side of the equation.
-41 + -20f + 41 + f2 = 0 + 41

Reorder the terms:
-41 + 41 + -20f + f2 = 0 + 41

Combine like terms: -41 + 41 = 0
0 + -20f + f2 = 0 + 41
-20f + f2 = 0 + 41

Combine like terms: 0 + 41 = 41
-20f + f2 = 41

The f term is -20f.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20f + 100 + f2 = 41 + 100

Reorder the terms:
100 + -20f + f2 = 41 + 100

Combine like terms: 41 + 100 = 141
100 + -20f + f2 = 141

Factor a perfect square on the left side:
(f + -10)(f + -10) = 141

Calculate the square root of the right side: 11.874342087

Break this problem into two subproblems by setting 
(f + -10) equal to 11.874342087 and -11.874342087.


Subproblem 1
f + -10 = 11.874342087

Simplifying
f + -10 = 11.874342087

Reorder the terms:
-10 + f = 11.874342087

Solving
-10 + f = 11.874342087

Solving for variable 'f'.

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

Add '10' to each side of the equation.
-10 + 10 + f = 11.874342087 + 10

Combine like terms: -10 + 10 = 0
0 + f = 11.874342087 + 10
f = 11.874342087 + 10

Combine like terms: 11.874342087 + 10 = 21.874342087
f = 21.874342087

Simplifying
f = 21.874342087


Subproblem 2
f + -10 = -11.874342087

Simplifying
f + -10 = -11.874342087

Reorder the terms:
-10 + f = -11.874342087

Solving
-10 + f = -11.874342087

Solving for variable 'f'.

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

Add '10' to each side of the equation.
-10 + 10 + f = -11.874342087 + 10

Combine like terms: -10 + 10 = 0
0 + f = -11.874342087 + 10
f = -11.874342087 + 10

Combine like terms: -11.874342087 + 10 = -1.874342087
f = -1.874342087

Simplifying
f = -1.874342087


Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {21.874342087, -1.874342087}

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