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Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?
- a)Four left half plane (LHP) roots
- b)One right half plane (RHP) root, one LHP root and two roots on the imaginary axis
- c)Two RHP roots and two LHP roots
- d)All four roots are on the imaginary axis
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Which one of the following options correctly describes the locations o...
The roots of the equation s^4 + s^2 + 1 = 0 on the complex plane can be described as follows:
Explanation:
To find the roots of the equation s^4 + s^2 + 1 = 0, we can rearrange it as follows:
s^4 + s^2 + 1 = 0
(s^2 + 1)^2 - s^2 = 0
(s^2 + 1 + s)(s^2 + 1 - s) = 0
This equation can be further simplified into two quadratic equations:
s^2 + 1 + s = 0 (Equation 1)
s^2 + 1 - s = 0 (Equation 2)
Roots of Equation 1:
To find the roots of Equation 1, we can use the quadratic formula:
s = (-b ± √(b^2 - 4ac)) / (2a)
For Equation 1, a = 1, b = 1, and c = 1. Substituting these values into the quadratic formula, we get:
s = (-1 ± √(1 - 4(1)(1))) / (2(1))
s = (-1 ± √(-3)) / 2
Since the discriminant (√(-3)) is imaginary, the roots of Equation 1 will also be imaginary. Therefore, Equation 1 has two roots on the imaginary axis.
Roots of Equation 2:
Similarly, using the quadratic formula for Equation 2, we get:
s = (1 ± √(1 - 4(1)(1))) / (2(1))
s = (1 ± √(-3)) / 2
Again, since the discriminant (√(-3)) is imaginary, the roots of Equation 2 will also be imaginary. Therefore, Equation 2 has two roots on the imaginary axis.
Conclusion:
In total, the equation s^4 + s^2 + 1 = 0 has four roots on the complex plane. Two of these roots are on the right half plane (RHP) and the other two roots are on the left half plane (LHP).
Explanation:
To find the roots of the equation s^4 + s^2 + 1 = 0, we can rearrange it as follows:
s^4 + s^2 + 1 = 0
(s^2 + 1)^2 - s^2 = 0
(s^2 + 1 + s)(s^2 + 1 - s) = 0
This equation can be further simplified into two quadratic equations:
s^2 + 1 + s = 0 (Equation 1)
s^2 + 1 - s = 0 (Equation 2)
Roots of Equation 1:
To find the roots of Equation 1, we can use the quadratic formula:
s = (-b ± √(b^2 - 4ac)) / (2a)
For Equation 1, a = 1, b = 1, and c = 1. Substituting these values into the quadratic formula, we get:
s = (-1 ± √(1 - 4(1)(1))) / (2(1))
s = (-1 ± √(-3)) / 2
Since the discriminant (√(-3)) is imaginary, the roots of Equation 1 will also be imaginary. Therefore, Equation 1 has two roots on the imaginary axis.
Roots of Equation 2:
Similarly, using the quadratic formula for Equation 2, we get:
s = (1 ± √(1 - 4(1)(1))) / (2(1))
s = (1 ± √(-3)) / 2
Again, since the discriminant (√(-3)) is imaginary, the roots of Equation 2 will also be imaginary. Therefore, Equation 2 has two roots on the imaginary axis.
Conclusion:
In total, the equation s^4 + s^2 + 1 = 0 has four roots on the complex plane. Two of these roots are on the right half plane (RHP) and the other two roots are on the left half plane (LHP).
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Community Answer
Which one of the following options correctly describes the locations o...
CE: s4 + 0s3 + 1s2 + 0s + 1
Routh array
We have row zero at s3 row
Solving the auxiliary equation, we get:
s4 + s2 + 1 = 0
By differentiating, we get:
4s3 + 2s = 0
The Routh array is modified as shown above.
Observations:
The row of zero indicates symmetric roots about the origin.
2 sign changes below row of zero indicate 2 poles in the right half of the s-plane.
∴ Two poles are on the right side and 2 poles symmetrically lying on left-half.
Routh array
We have row zero at s3 row
Solving the auxiliary equation, we get:
s4 + s2 + 1 = 0
By differentiating, we get:
4s3 + 2s = 0
The Routh array is modified as shown above.
Observations:
The row of zero indicates symmetric roots about the origin.
2 sign changes below row of zero indicate 2 poles in the right half of the s-plane.
∴ Two poles are on the right side and 2 poles symmetrically lying on left-half.
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Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer?
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Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer?.
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