z^4=z^3-42z^2

Solution for z^4=z^3-42z^2 equation:

Simplifying
z4 = z3 + -42z2

Reorder the terms:
z4 = -42z2 + z3

Solving
z4 = -42z2 + z3

Solving for variable 'z'.

Reorder the terms:
42z2 + -1z3 + z4 = -42z2 + z3 + 42z2 + -1z3

Reorder the terms:
42z2 + -1z3 + z4 = -42z2 + 42z2 + z3 + -1z3

Combine like terms: -42z2 + 42z2 = 0
42z2 + -1z3 + z4 = 0 + z3 + -1z3
42z2 + -1z3 + z4 = z3 + -1z3

Combine like terms: z3 + -1z3 = 0
42z2 + -1z3 + z4 = 0

Factor out the Greatest Common Factor (GCF), 'z2'.
z2(42 + -1z + z2) = 0

Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve:

Simplifying
z2 = 0

Solving
z2 = 0

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

Simplifying
z2 = 0

Take the square root of each side:
z = {0}
Subproblem 2
Set the factor '(42 + -1z + z2)' equal to zero and attempt to solve:

Simplifying
42 + -1z + z2 = 0

Solving
42 + -1z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-42' to each side of the equation.
42 + -1z + -42 + z2 = 0 + -42

Reorder the terms:
42 + -42 + -1z + z2 = 0 + -42

Combine like terms: 42 + -42 = 0
0 + -1z + z2 = 0 + -42
-1z + z2 = 0 + -42

Combine like terms: 0 + -42 = -42
-1z + z2 = -42

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = -42 + 0.25

Reorder the terms:
0.25 + -1z + z2 = -42 + 0.25

Combine like terms: -42 + 0.25 = -41.75
0.25 + -1z + z2 = -41.75

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = -41.75

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
z = {0}