20f^2-21f-4=0

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Solution for 20f^2-21f-4=0 equation:


Simplifying
20f2 + -21f + -4 = 0

Reorder the terms:
-4 + -21f + 20f2 = 0

Solving
-4 + -21f + 20f2 = 0

Solving for variable 'f'.

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

Divide each side by '20'.
-0.2 + -1.05f + f2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -1.05f + 0.2 + f2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -1.05f + f2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -1.05f + f2 = 0 + 0.2
-1.05f + f2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-1.05f + f2 = 0.2

The f term is -1.05f.  Take half its coefficient (-0.525).
Square it (0.275625) and add it to both sides.

Add '0.275625' to each side of the equation.
-1.05f + 0.275625 + f2 = 0.2 + 0.275625

Reorder the terms:
0.275625 + -1.05f + f2 = 0.2 + 0.275625

Combine like terms: 0.2 + 0.275625 = 0.475625
0.275625 + -1.05f + f2 = 0.475625

Factor a perfect square on the left side:
(f + -0.525)(f + -0.525) = 0.475625

Calculate the square root of the right side: 0.689655711

Break this problem into two subproblems by setting 
(f + -0.525) equal to 0.689655711 and -0.689655711.

Subproblem 1

f + -0.525 = 0.689655711 Simplifying f + -0.525 = 0.689655711 Reorder the terms: -0.525 + f = 0.689655711 Solving -0.525 + f = 0.689655711 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + f = 0.689655711 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + f = 0.689655711 + 0.525 f = 0.689655711 + 0.525 Combine like terms: 0.689655711 + 0.525 = 1.214655711 f = 1.214655711 Simplifying f = 1.214655711

Subproblem 2

f + -0.525 = -0.689655711 Simplifying f + -0.525 = -0.689655711 Reorder the terms: -0.525 + f = -0.689655711 Solving -0.525 + f = -0.689655711 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + f = -0.689655711 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + f = -0.689655711 + 0.525 f = -0.689655711 + 0.525 Combine like terms: -0.689655711 + 0.525 = -0.164655711 f = -0.164655711 Simplifying f = -0.164655711

Solution

The solution to the problem is based on the solutions from the subproblems. f = {1.214655711, -0.164655711}

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