r^4+r^3s+r^2s^2=0

Solution for r^4+r^3s+r^2s^2=0 equation:

Simplifying
r4 + r3s + r2s2 = 0

Reorder the terms:
r2s2 + r3s + r4 = 0

Solving
r2s2 + r3s + r4 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), 'r2'.
r2(s2 + rs + r2) = 0

Subproblem 1
Set the factor 'r2' equal to zero and attempt to solve:

Simplifying
r2 = 0

Solving
r2 = 0

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

Simplifying
r2 = 0

Take the square root of each side:
r = {0}
Subproblem 2
Set the factor '(s2 + rs + r2)' equal to zero and attempt to solve:

Simplifying
s2 + rs + r2 = 0

Reorder the terms:
rs + r2 + s2 = 0

Solving
rs + r2 + s2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1s2' to each side of the equation.
rs + r2 + s2 + -1s2 = 0 + -1s2

Combine like terms: s2 + -1s2 = 0
rs + r2 + 0 = 0 + -1s2
rs + r2 = 0 + -1s2
Remove the zero:
rs + r2 = -1s2

The r term is rs.  Take half its coefficient (0.5s).
Square it (0.25s2) and add it to both sides.

Add '0.25s2' to each side of the equation.
rs + r2 + 0.25s2 = -1s2 + 0.25s2

Combine like terms: -1s2 + 0.25s2 = -0.75s2
rs + r2 + 0.25s2 = -0.75s2

Factor a perfect square on the left side:
(r + 0.5s)(r + 0.5s) = -0.75s2

Can't calculate square root of the right side.

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
r = {0}