Let's solve the equation [tex]\(x^4 - x = 0\)[/tex] step-by-step.
1. Factor the given equation:
[tex]\[
x^4 - x = 0
\][/tex]
We can factor out [tex]\(x\)[/tex] from the equation:
[tex]\[
x(x^3 - 1) = 0
\][/tex]
2. Solve for the individual factors:
This gives us two separate equations to solve:
[tex]\[
x = 0 \quad \text{or} \quad x^3 - 1 = 0
\][/tex]
3. Solve [tex]\(x = 0\)[/tex]:
One of the solutions is straightforward:
[tex]\[
x = 0
\][/tex]
4. Solve [tex]\(x^3 - 1 = 0\)[/tex]:
[tex]\[
x^3 - 1 = 0 \implies x^3 = 1
\][/tex]
To solve this, we find the cube roots of 1. The equation [tex]\(x^3 = 1\)[/tex] has three roots, one real and two complex:
- The real root:
[tex]\[
x = 1
\][/tex]
- The complex roots: To find the complex roots, we express 1 in its polar form. In polar form, 1 can be written as:
[tex]\[
1 = e^{2k\pi i} \quad \text{for} \quad k = 0, 1, \text{and} 2
\][/tex]
The cube roots of unity are given by:
[tex]\[
x = e^{2k\pi i / 3} \quad \text{for} \quad k = 0, 1, \text{and} 2
\][/tex]
Substituting [tex]\(k = 0, 1,\)[/tex] and [tex]\(2\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
x = e^{0} = 1
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
x = e^{2\pi i / 3} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
x = e^{4\pi i / 3} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i
\][/tex]
5. List all solutions:
Combining all solutions, we get:
[tex]\[
x = 0, \quad x = 1, \quad x = -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad x = -\frac{1}{2} - \frac{\sqrt{3}}{2}i
\][/tex]
Putting it all together, the solutions to the equation [tex]\(x^4 - x = 0\)[/tex] are:
[tex]\[
0, 1, -0.5 + 2.1213203435596424i, -0.5 - 2.1213203435596424i
\][/tex]
