f^2+5f-19=0

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Solution for f^2+5f-19=0 equation:


Simplifying
f2 + 5f + -19 = 0

Reorder the terms:
-19 + 5f + f2 = 0

Solving
-19 + 5f + f2 = 0

Solving for variable 'f'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 5f + 19 + f2 = 0 + 19

Reorder the terms:
-19 + 19 + 5f + f2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 5f + f2 = 0 + 19
5f + f2 = 0 + 19

Combine like terms: 0 + 19 = 19
5f + f2 = 19

The f term is 5f.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5f + 6.25 + f2 = 19 + 6.25

Reorder the terms:
6.25 + 5f + f2 = 19 + 6.25

Combine like terms: 19 + 6.25 = 25.25
6.25 + 5f + f2 = 25.25

Factor a perfect square on the left side:
(f + 2.5)(f + 2.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(f + 2.5) equal to 5.024937811 and -5.024937811.

Subproblem 1

f + 2.5 = 5.024937811 Simplifying f + 2.5 = 5.024937811 Reorder the terms: 2.5 + f = 5.024937811 Solving 2.5 + f = 5.024937811 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + f = 5.024937811 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + f = 5.024937811 + -2.5 f = 5.024937811 + -2.5 Combine like terms: 5.024937811 + -2.5 = 2.524937811 f = 2.524937811 Simplifying f = 2.524937811

Subproblem 2

f + 2.5 = -5.024937811 Simplifying f + 2.5 = -5.024937811 Reorder the terms: 2.5 + f = -5.024937811 Solving 2.5 + f = -5.024937811 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + f = -5.024937811 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + f = -5.024937811 + -2.5 f = -5.024937811 + -2.5 Combine like terms: -5.024937811 + -2.5 = -7.524937811 f = -7.524937811 Simplifying f = -7.524937811

Solution

The solution to the problem is based on the solutions from the subproblems. f = {2.524937811, -7.524937811}

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