# Solve for w w^4-w=0

w4-w=0
Factor the left side of the equation.
Factor w out of w4-w.
Factor w out of w4.
w⋅w3-w=0
Factor w out of -w.
w⋅w3+w⋅-1=0
Factor w out of w⋅w3+w⋅-1.
w(w3-1)=0
w(w3-1)=0
Rewrite 1 as 13.
w(w3-13)=0
Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=w and b=1.
w((w-1)(w2+w⋅1+12))=0
Factor.
Simplify.
Multiply w by 1.
w((w-1)(w2+w+12))=0
One to any power is one.
w((w-1)(w2+w+1))=0
w((w-1)(w2+w+1))=0
Remove unnecessary parentheses.
w(w-1)(w2+w+1)=0
w(w-1)(w2+w+1)=0
w(w-1)(w2+w+1)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
w=0
w-1=0
w2+w+1=0
Set the first factor equal to 0.
w=0
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
w-1=0
Add 1 to both sides of the equation.
w=1
w=1
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
w2+w+1=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=1, and c=1 into the quadratic formula and solve for w.
-1±12-4⋅(1⋅1)2⋅1
Simplify.
Simplify the numerator.
One to any power is one.
w=-1±1-4⋅(1⋅1)2⋅1
Multiply 1 by 1.
w=-1±1-4⋅12⋅1
Multiply -4 by 1.
w=-1±1-42⋅1
Subtract 4 from 1.
w=-1±-32⋅1
Rewrite -3 as -1(3).
w=-1±-1⋅32⋅1
Rewrite -1(3) as -1⋅3.
w=-1±-1⋅32⋅1
Rewrite -1 as i.
w=-1±i32⋅1
w=-1±i32⋅1
Multiply 2 by 1.
w=-1±i32
w=-1±i32
The final answer is the combination of both solutions.
w=-1-i32,-1+i32
w=-1-i32,-1+i32
The final solution is all the values that make w(w-1)(w2+w+1)=0 true.
w=0,1,-1-i32,-1+i32
Solve for w w^4-w=0