f^2+10f=-1

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Solution for f^2+10f=-1 equation:


Simplifying
f2 + 10f = -1

Reorder the terms:
10f + f2 = -1

Solving
10f + f2 = -1

Solving for variable 'f'.

Reorder the terms:
1 + 10f + f2 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + 10f + f2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 10f + -1 + f2 = 0 + -1

Reorder the terms:
1 + -1 + 10f + f2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 10f + f2 = 0 + -1
10f + f2 = 0 + -1

Combine like terms: 0 + -1 = -1
10f + f2 = -1

The f term is 10f.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10f + 25 + f2 = -1 + 25

Reorder the terms:
25 + 10f + f2 = -1 + 25

Combine like terms: -1 + 25 = 24
25 + 10f + f2 = 24

Factor a perfect square on the left side:
(f + 5)(f + 5) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(f + 5) equal to 4.898979486 and -4.898979486.

Subproblem 1

f + 5 = 4.898979486 Simplifying f + 5 = 4.898979486 Reorder the terms: 5 + f = 4.898979486 Solving 5 + f = 4.898979486 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + f = 4.898979486 + -5 Combine like terms: 5 + -5 = 0 0 + f = 4.898979486 + -5 f = 4.898979486 + -5 Combine like terms: 4.898979486 + -5 = -0.101020514 f = -0.101020514 Simplifying f = -0.101020514

Subproblem 2

f + 5 = -4.898979486 Simplifying f + 5 = -4.898979486 Reorder the terms: 5 + f = -4.898979486 Solving 5 + f = -4.898979486 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + f = -4.898979486 + -5 Combine like terms: 5 + -5 = 0 0 + f = -4.898979486 + -5 f = -4.898979486 + -5 Combine like terms: -4.898979486 + -5 = -9.898979486 f = -9.898979486 Simplifying f = -9.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. f = {-0.101020514, -9.898979486}

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