-5f^2-7f-1=0

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Solution for -5f^2-7f-1=0 equation: 


Simplifying
-5f2 + -7f + -1 = 0

Reorder the terms:
-1 + -7f + -5f2 = 0

Solving
-1 + -7f + -5f2 = 0

Solving for variable 'f'.

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 7f + 5f2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(1 + 7f + 5f2)' equal to zero and attempt to solve:

Simplifying
1 + 7f + 5f2 = 0

Solving
1 + 7f + 5f2 = 0

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

Divide each side by '5'.
0.2 + 1.4f + f2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 1.4f + -0.2 + f2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 1.4f + f2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 1.4f + f2 = 0 + -0.2
1.4f + f2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
1.4f + f2 = -0.2

The f term is 1.4f.  Take half its coefficient (0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
1.4f + 0.49 + f2 = -0.2 + 0.49

Reorder the terms:
0.49 + 1.4f + f2 = -0.2 + 0.49

Combine like terms: -0.2 + 0.49 = 0.29
0.49 + 1.4f + f2 = 0.29

Factor a perfect square on the left side:
(f + 0.7)(f + 0.7) = 0.29

Calculate the square root of the right side: 0.538516481

Break this problem into two subproblems by setting 
(f + 0.7) equal to 0.538516481 and -0.538516481.


Subproblem 1
f + 0.7 = 0.538516481

Simplifying
f + 0.7 = 0.538516481

Reorder the terms:
0.7 + f = 0.538516481

Solving
0.7 + f = 0.538516481

Solving for variable 'f'.

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

Add '-0.7' to each side of the equation.
0.7 + -0.7 + f = 0.538516481 + -0.7

Combine like terms: 0.7 + -0.7 = 0.0
0.0 + f = 0.538516481 + -0.7
f = 0.538516481 + -0.7

Combine like terms: 0.538516481 + -0.7 = -0.161483519
f = -0.161483519

Simplifying
f = -0.161483519


Subproblem 2
f + 0.7 = -0.538516481

Simplifying
f + 0.7 = -0.538516481

Reorder the terms:
0.7 + f = -0.538516481

Solving
0.7 + f = -0.538516481

Solving for variable 'f'.

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

Add '-0.7' to each side of the equation.
0.7 + -0.7 + f = -0.538516481 + -0.7

Combine like terms: 0.7 + -0.7 = 0.0
0.0 + f = -0.538516481 + -0.7
f = -0.538516481 + -0.7

Combine like terms: -0.538516481 + -0.7 = -1.238516481
f = -1.238516481

Simplifying
f = -1.238516481


Solution
The solution to the problem is based on the solutions
from the subproblems.
f = {-0.161483519, -1.238516481}
Solution
f = {-0.161483519, -1.238516481}

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