Question
Recognize which of the following statements is true.(1 point)ResponsesWhen k is positive, x3=k has one positive solution and one negative solution.When k is positive, x cubed equals k has one positive solution and one negative solution.When k is positive, x2=k has one solution, which is positive.When k is positive, x squared equals k has one solution, which is positive.When k is negative, x2=k has one solution, which is negative.When k is negative, x squared equals k has one solution, which is negative.When k is negative, x3=k has one solution, which is negative.
Answers
GPT-4o mini
Let's analyze each statement one by one regarding the equations provided:
1. **When k is positive, \(x^3 = k\) has one positive solution and one negative solution.**
- **False**: \(x^3 = k\) will have one positive solution (since the cube root of a positive number is positive) and no negative solution.
2. **When k is positive, \(x^3 = k\) has one positive solution and one negative solution.**
- **False**: This is identical to the first statement and is incorrect for the same reason.
3. **When k is positive, \(x^2 = k\) has one solution, which is positive.**
- **False**: \(x^2 = k\) has two solutions: \(x = \sqrt{k}\) (positive) and \(x = -\sqrt{k}\) (negative).
4. **When k is positive, \(x^2 = k\) has one solution, which is positive.**
- **False**: This is the same as the third statement and is incorrect for the same reason.
5. **When k is negative, \(x^2 = k\) has one solution, which is negative.**
- **False**: Since \(x^2\) is non-negative for all real \(x\), there are no real solutions when \(k\) is negative.
6. **When k is negative, \(x^2 = k\) has one solution, which is negative.**
- **False**: This is the same as the fifth statement and is incorrect for the same reason.
7. **When k is negative, \(x^3 = k\) has one solution, which is negative.**
- **True**: The equation \(x^3 = k\) where \(k\) is negative will have one real solution that is negative (the cube root of a negative number is negative).
Based on this analysis, the true statement is:
**"When k is negative, \(x^3 = k\) has one solution, which is negative."**
1. **When k is positive, \(x^3 = k\) has one positive solution and one negative solution.**
- **False**: \(x^3 = k\) will have one positive solution (since the cube root of a positive number is positive) and no negative solution.
2. **When k is positive, \(x^3 = k\) has one positive solution and one negative solution.**
- **False**: This is identical to the first statement and is incorrect for the same reason.
3. **When k is positive, \(x^2 = k\) has one solution, which is positive.**
- **False**: \(x^2 = k\) has two solutions: \(x = \sqrt{k}\) (positive) and \(x = -\sqrt{k}\) (negative).
4. **When k is positive, \(x^2 = k\) has one solution, which is positive.**
- **False**: This is the same as the third statement and is incorrect for the same reason.
5. **When k is negative, \(x^2 = k\) has one solution, which is negative.**
- **False**: Since \(x^2\) is non-negative for all real \(x\), there are no real solutions when \(k\) is negative.
6. **When k is negative, \(x^2 = k\) has one solution, which is negative.**
- **False**: This is the same as the fifth statement and is incorrect for the same reason.
7. **When k is negative, \(x^3 = k\) has one solution, which is negative.**
- **True**: The equation \(x^3 = k\) where \(k\) is negative will have one real solution that is negative (the cube root of a negative number is negative).
Based on this analysis, the true statement is:
**"When k is negative, \(x^3 = k\) has one solution, which is negative."**