5f^2+4fg-g^2=0

Solution for 5f^2+4fg-g^2=0 equation:

Simplifying
5f2 + 4fg + -1g2 = 0

Reorder the terms:
4fg + 5f2 + -1g2 = 0

Solving
4fg + 5f2 + -1g2 = 0

Solving for variable 'f'.

Factor a trinomial.
(5f + -1g)(f + g) = 0

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

Simplifying
5f + -1g = 0

Solving
5f + -1g = 0

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

Add 'g' to each side of the equation.
5f + -1g + g = 0 + g

Combine like terms: -1g + g = 0
5f + 0 = 0 + g
5f = 0 + g
Remove the zero:
5f = g

Divide each side by '5'.
f = 0.2g

Simplifying
f = 0.2g
Subproblem 2
Set the factor '(f + g)' equal to zero and attempt to solve:

Simplifying
f + g = 0

Solving
f + g = 0

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

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

Combine like terms: g + -1g = 0
f + 0 = 0 + -1g
f = 0 + -1g
Remove the zero:
f = -1g

Simplifying
f = -1g
Solution
f = {0.2g, -1g}