Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.