4f^2+104f+196=0

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Solution for 4f^2+104f+196=0 equation: 


Simplifying
4f2 + 104f + 196 = 0

Reorder the terms:
196 + 104f + 4f2 = 0

Solving
196 + 104f + 4f2 = 0

Solving for variable 'f'.

Factor out the Greatest Common Factor (GCF), '4'.
4(49 + 26f + f2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(49 + 26f + f2)' equal to zero and attempt to solve:

Simplifying
49 + 26f + f2 = 0

Solving
49 + 26f + f2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-49' to each side of the equation.
49 + 26f + -49 + f2 = 0 + -49

Reorder the terms:
49 + -49 + 26f + f2 = 0 + -49

Combine like terms: 49 + -49 = 0
0 + 26f + f2 = 0 + -49
26f + f2 = 0 + -49

Combine like terms: 0 + -49 = -49
26f + f2 = -49

The f term is 26f.  Take half its coefficient (13).
Square it (169) and add it to both sides.

Add '169' to each side of the equation.
26f + 169 + f2 = -49 + 169

Reorder the terms:
169 + 26f + f2 = -49 + 169

Combine like terms: -49 + 169 = 120
169 + 26f + f2 = 120

Factor a perfect square on the left side:
(f + 13)(f + 13) = 120

Calculate the square root of the right side: 10.95445115

Break this problem into two subproblems by setting 
(f + 13) equal to 10.95445115 and -10.95445115.


Subproblem 1
f + 13 = 10.95445115

Simplifying
f + 13 = 10.95445115

Reorder the terms:
13 + f = 10.95445115

Solving
13 + f = 10.95445115

Solving for variable 'f'.

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

Add '-13' to each side of the equation.
13 + -13 + f = 10.95445115 + -13

Combine like terms: 13 + -13 = 0
0 + f = 10.95445115 + -13
f = 10.95445115 + -13

Combine like terms: 10.95445115 + -13 = -2.04554885
f = -2.04554885

Simplifying
f = -2.04554885


Subproblem 2
f + 13 = -10.95445115

Simplifying
f + 13 = -10.95445115

Reorder the terms:
13 + f = -10.95445115

Solving
13 + f = -10.95445115

Solving for variable 'f'.

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

Add '-13' to each side of the equation.
13 + -13 + f = -10.95445115 + -13

Combine like terms: 13 + -13 = 0
0 + f = -10.95445115 + -13
f = -10.95445115 + -13

Combine like terms: -10.95445115 + -13 = -23.95445115
f = -23.95445115

Simplifying
f = -23.95445115


Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {-2.04554885, -23.95445115}
Solution
f = {-2.04554885, -23.95445115}

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